Suppose that I have a linear model with autocorrelated errors. Is there any results telling me that if I assume iid errors I overestimate or underestimate my standard errors ?
There are no definite results (actually there are but it depends on the autocorrelation structure) for this and the reason is that if you know the correlation structure sufficiently, you can also correct for it and regain BLUE. So with the information you seek you no longer need to mitigate the error of regular SE, as you can use more appropriate measures to get a precise estimate.
The easiest option is just to use Newey/West estimates (in R, you can use the vcovHAC option of the sandwhich package). These are consistent (for large samples). This is the "easy way out".
If your sample is smaller or you can estimate a constant correlation structure, you can use (F)GLS estimation. Search for (iterative) Cochrane/Orcutt and Prais/Winston for two options on how to do this. Estimate iteratively with FGLS if you do know the structure, but not the parameter of the autocorrelation.
You can get some insight by using a Breusch-Godfrey test. This should give you some idea if there is a well defined autocorrelation structure.
Be aware though that FGLS still needs contemporary exogenity of the regressors!