You add an extra covariate to the design matrix when you expect that it will explain some of the variance in gene expression. In general, i think this IS the case for matched samples.
Including additional explanatory co-variates means you have more coefficients to fit in your model. As we know, the more coefficients you need to fit, the more samples you need to do it, because these coefficients increase the number of degrees of freedom used. This reduces your power. However, adding coefficients reduces the variance for the coefficient you are interested in measuring. Which increases power. Generally, your overall power will be increased if the co-variate explains a substantial portion of the variance, and decreaed if it doesn't.
One way to check would be to look at a PCA. Do the points cluster in one of the first few PCs according to the levels of your matching criteria?
My intution is that you should see if things cluster in PCA, and then run tests on that coefficient looking for differences, because that would be post-hoc double dipping. But I think (and a statistician might correct me if i'm wrong), this is okay if you are fitting this coefficient in order to ignore it.