Are there algorithms for computing "running" linear or logistic regression parameters? A paper "Accurately computing running variance" at http://www.johndcook.com/standard_deviation.html
shows how to compute running mean, variance and standard deviations.
Are there algorithms where the parameters of a linear or logistic regression model can be similarly "dynamically" updated as each new training record is provided?
 A: Adding to tdc's answer, there are no known methods to compute exact estimates of the coefficients at any point in time with just constant time per iteration. However, there are some alternatives which are reasonable and interesting.
The first model to look at is the online learning setting. In this setting, the world first announces a value of x, your algorithm predicts a value for y, the world announces the true value y', and your algorithm suffers a loss l(y,y'). For this setting it is known that simple algorithms (gradient descent and exponentiated gradient, among others) achieve sublinear regret. This means that as you see more examples the number of extra mistakes your algorithm makes (when compared to the best possible linear predictor) does not grow with the number of examples. This works even in adversarial settings. There is a good paper explaining one popular strategy to prove these regret bounds. Shai Shalev-Schwartz's lecture notes are also useful.
There is an extension of the online learning setting called the bandit setting where your algorithm is only given a number representing how wrong it was (and no pointer to the right answer). Impressively, many results from online learning carry over to this setting, except here one is forced to explore as well as exploit, which leads to all sorts of interesting challenges.
A: You can use some standard Kalman Filter package in R for this - sspir, dlm, kfas, etc. I feel that KF is a much more developed area than online-learning, so it may be more practical. You may use a model
$$y_t = \beta_t\cdot x_t + \varepsilon_t, \\ \beta_t = \beta_{t-1}+ \eta_t$$ to allow your regression coefficients to slowly vary with time and KF will re-estimate them on each step (with constant time cost) based on most recent data. Alternatively, you can set them constant $$\beta_t = \beta_{t-1}$$ and KF will still re-estimate them on each step but this time assuming they are constant and just incorporating new observed data to produce better and better estimates of same coefficients values. 
You can formulate similar model for logistic regression, $$y_t = logit(\beta_t\cdot x_t + \varepsilon_t), \\ \beta_t = \beta_{t-1}+ \eta_t$$ as it will be non-linear, you will need to use non-linear filtering method from above packages - EKF or UKF.
A: Other answers have pointed to the world of machine learning, and that is certainly one place where this problem has been addressed.  
However, another approach that may be better suited to your needs is the use of the QR factorization with with low rank updates.  Approaches to doing this and using it to solve least squares problems are given in:
Updating the QR factorization and the least squares problem by Hammerling and Lucas.
A: The linear régression coefficients of $y = ax + b$ are $a = cov(x,y)/var(x)$ and $b = mean(y) - a \cdot mean(x)$.
So all you really need is an incremental method to compute $cov(x,y)$. From this value and the variance of $x$ and the mean of both $y$ and $x$ you can compute the parameters $a$ and $b$.
As you will see in the pseudo code given below incremental computation of $cov(x,y)$ is very similar to incremental computation of $var(x)$. This shouldn't be a surprise because $var(x) = cov(x,x)$. 
Here is the pseudo code you are probably looking for:
init(): meanX = 0, meanY = 0, varX = 0, covXY = 0, n = 0

update(x,y):
n += 1
dx = x - meanX
dy = y - meanY
varX += (((n-1)/n)*dx*dx - varX)/n
covXY += (((n-1)/n)*dx*dy - covXY)/n
meanX += dx/n
meanY += dy/n

getA(): return covXY/varX
getB(): return meanY - getA()*meanX

I found this question while searching for an equivalent algorithm incrementally computing a multi variate regression as $R = (X'X)^{-1}X'Y $ so that $ XR = Y+\epsilon $
A: This is to add to @chmike answer.
The method appears to be similar to B. P. Welford’s online algorithm for standard deviation which also calculates the mean.  John Cook gives a good explanation here. Tony Finch in 2009 provides a method for an exponential moving average and standard deviation:
diff := x – mean 
incr := alpha * diff 
mean := mean + incr 
variance := (1 - alpha) * (variance + diff * incr)

Peering at the previously posted answer and expanding upon it to include a exponential moving window:
init(): 
    meanX = 0, meanY = 0, varX = 0, covXY = 0, n = 0,
    meanXY = 0, varY = 0, desiredAlpha=0.01 #additional variables for correlation

update(x,y):
    n += 1
    alpha=max(desiredAlpha,1/n) #to handle initial conditions

    dx = x - meanX
    dy = y - meanY
    dxy = (x*y) - meanXY #needed for cor

    varX += ((1-alpha)*dx*dx - varX)*alpha
    varY += ((1-alpha)*dy*dy - varY)*alpha #needed for corXY
    covXY += ((1-alpha)*dx*dy - covXY)*alpha

    #alternate method: varX = (1-alpha)*(varX+dx*dx*alpha)
    #alternate method: varY = (1-alpha)*(varY+dy*dy*alpha) #needed for corXY
    #alternate method: covXY = (1-alpha)*(covXY+dx*dy*alpha)

    meanX += dx * alpha
    meanY += dy * alpha
    meanXY += dxy  * alpha

getA(): return covXY/varX
getB(): return meanY - getA()*meanX
corXY(): return (meanXY - meanX * meanY) / ( sqrt(varX) * sqrt(varY) )

In the above "code", desiredAlpha could be set to 0 and if so, the code would operate without exponential weighting. It can be suggested to set desiredAlpha to 1/desiredWindowSize as suggested by Modified_moving_average for a moving window size.
Side question: of the alternative calculations above, any comments on which is better from a precision standpoint?
References:
chmike (2013) https://stats.stackexchange.com/a/79845/70282
Cook, John (n.d.) Accurately computing running variance http://www.johndcook.com/blog/standard_deviation/
Finch, Tony. (2009) Incremental calculation of weighted mean and variance. https://fanf2.user.srcf.net/hermes/doc/antiforgery/stats.pdf
Wikipedia. (n.d) Welford’s online algorithm https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance#Online_algorithm
A: For your two specific examples:
Linear Regression
The paper "Online Linear Regression and Its Application to Model-Based Reinforcement Learning" by Alexander Strehl and Michael Littman describes an algorithm called "KWIK Linear Regression" (see algorithm 1) which provides an approximation to the linear regression solution using incremental updates. Note that this is not regularised (i.e. it is not Ridge Regression). I'm pretty sure that the method of Strehl & Littman cannot extend to that setting.
Logistic Regression
This thread sheds some light on the matter. Quoting:

Even without a regularization constraint, logistic regression is a nonlinear optimization problem. Already this does not have an analytic solution, which is usually a prerequisite to deriving an update solution. With a regularization constraint, it becomes a constrained optimization problem. This introduces a whole new set of non-analytic complications on top of the ones that the unconstrained problem already had.

There are however other online (or incremental) methods for regression that you might want to look at, for example Locally Weighted Projection Regression (LWPR)
A: As a general principle:
0) you keep the sufficient statistics and the current ML estimates
1) when you get new data, update the sufficient statistics and the estimates
2) When you don't have sufficient statistics you'll need to use all of the data.
3) Typically you don't have closed-form solutions; use the previous MLEs as the starting point, use some convenient optimization method to find the new optimum from there. You may need to experiment a bit to find which approaches make the best tradeoffs for your particular kinds of problem instances.
If your problem has a special structure, you can probably exploit it.
A couple of potential references that may or may not have some value:
McMahan, H. B. and M. Streeter (2012),
Open Problem: Better Bounds for Online Logistic Regression,
JMLR: Workshop and Conference Proceedings, vol 23, 44.1–44.3  
Penny, W.D. and S.J. Roberts (1999),
Dynamic Logistic Regression,
Proceedings IJCNN '99  
