Coefficient estimated with a binary predictor $\in \{0,1\}$, but making predictions with values between $0$ and $1$ - is this OK? Let's say I have a variable $x_d$ that, in the estimation data, is a simple indicator ($x_d \in \left\{0,1\right\}$).  I estimate a coefficient for it, $\beta_d$, along with several other coefficients for variables that could be continuous, more dummies, etc.
Now someone wants to use these coefficients in a forecast. For their input data, they would like to treat $x_d$ not as an indicator, but rather as a continuous variable. That is, while the estimation data represented this quantity as an either/or, the forecast data represents it more as a matter of degree.  In this forecast data, I would hope that my colleague would constrain the value of $x_d$ between 0 and 1, but we have not yet discussed that.
Edit
$x_d$ describes whether a loan came from a very specific lender.  So in reality, the value really is only 0 or 1.  In the forecast, my colleague wants to treat that variable as a continuous measure of similarity to that lender, i.e. how much like that lender is a loan’s source, with 0 meaning not at all, and 1 meaning the same lender.  Is it a bad thing to alter the meaning in that way?  I’m especially concerned about how it affects the meanings of the other coefficients, which were estimated alongside an indicator and are now being used alongside a continuous variable.
Are there any issues when a coefficient is estimated with an indicator/dummy, but then used in a forecast with a continuous variable?
 A: After clarification in a comment discussion, it appears that the question is about a situation where there is a linear model relating a continuous response variable $Y_i$ and a binary predictor $X_i \in \{0,1\}$, and the OP wants to know whether or not it is defensible to interpolate by plugging in values for $X_i$ that are $\in (0,1)$ and assuming everything is "OK". That answer is that this is only defensible if you assume linearity of $E(Y_i | X_i)$ over that interval. 
To see why this is true, consider fitting these data with a simple regression model: 
$$ Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i $$ 
where $\varepsilon_i$ is a mean $0$ error term. Under this model, $E(Y_i | X_i = 1) = \beta_0 + \beta_1$ and $E(Y_i | X_i = 0) = \beta_0$. When you only use data such that each $X_i \in \{0,1\}$ to fit this model, the coefficient estimates will be exactly
$$ \hat \beta_0 = \overline{Y_0}$$
and
$$ \hat \beta_1 = \overline{Y_1} - \overline{Y_0}$$
where $\overline{Y_0}$ is the sample mean of the $Y_i$'s when $X_i = 0$ and $\overline{Y_1}$ is the sample mean of the $Y_i$'s when $X_i = 1$. Therefore, the estimated mean for any intermediate value of $X_i$, call it $x$, is 
$$ \overline{Y_0} + (\overline{Y_1} - \overline{Y_0})x = \overline{Y_0}(1-x) + \overline{Y_1}x$$
That is, a function which linearly goes from $\overline{Y_0}$ to $\overline{Y_1}$ as $x$ moves from $0$ to $1$. If the true relationship is not this way, then the predictions made will be very wrong. 
For example, suppose the true relationship was a step function: 
$$E(Y_i | X_i) = \beta_0 + \beta_1 \mathcal{I}(X_i \geq 1/2) $$
where $\mathcal{I}(X_i \geq 1/2) = 1$ if $X_i \geq 1/2$ and 0 otherwise. This would be compatible with the linear model specified if all you observe are $X_i = 0,1$ but once you start to interpolate that linear relationship, you will get very wrong predictions. At $X_i = 1/2$, the true mean is $\beta_0 + \beta_1$ but the linear model's estimate would be $\beta_0 + \beta_1/2$ (Yikes!). Other examples can be given where the linear model is correct for the binary data, but makes this disagreement arbitrarily large once you interpolate.
A: Of course I agree with Macro here.  These are two entirely different models.  The coefficient for an indicator variable would bear no relation to what the coefficient should have been had you fit the model with a continuous variable.  If you have data on the continuous variable you would be better off fitting the model to that.  By dichotomizing the variable you throw out information in your data.
