Multicollinearity in case of nonstationary variables I am using dynamic OLS regression and fully modified OLS regression (in eViews) to estimate long run relationships' coefficients; my variables are all I(1).
Do I need to check for multicollinearity before running the estimation?
 A: By the sounds of it I think you will have to carry out some other tests before you get worried about multicollinearity.
Firstly, regression involving I(1) variables will have an R-squared increasing to 1 as the number of observations, T, increases. i.e. correlation will either go to 1 or -1, regardless of there being any relationship between the variables.
See: http://faculty.washington.edu/ezivot/econ584/notes/cointegration.pdf
(Zivot also has other related topics, which you may find useful)
When all elements of your correlation matrix (which you need to estimate implicitly for OLS) tend to 1 you will see what could look like multicollinearity. I know you said you are using dynamic regressions (which I assume you have a rolling window for estimations i.e. you fix T), that won't change the fact that you could be falling into a spurious regression.
I think using dynamic cointegration would be a better approach here, and you would not have to do much in terms of your model i.e. you could use OLS to estimate the cointegrating vector (your slope coefficients) via Engle-Granger and then carry out a Dickey-Fuller test for stationarity on the residuals at each regression you do. If your residuals are stationary (i.e. I(0)) at some level of significance you are likely to have a legitimate relationship between the variables. Make sure though that all your variables are I(1).
What's also nice about this approach is that you could modify it to an error correction model for prediction or fit mean reverting models to the residual (Ornstein-Uhlenbeck or Autoregressions) also for prediction.
