Does it make sense to build a residual network with only fully connected layers (instedad of convolutional layers)?
As Tapio, I also disagree with Giuseppe's conclusion. Residual layers are said to help improving performance in multiple ways: They let the gradient flow better, they might help in localization etc.  My guess is that some of these advantages like gradient flow, hold also for networks consisting of fully connected layers.
Other ideas like saying we learn residuals F(X)-X, where F is a residual block are more questionable due to absence of spatial correlation. That is, for CNNs, where residual connections are mostly used, we have some form of locality, that is if you have the feature map X of some layer (you can also think of X as the input) and the output of a residual block F(X) then both X and F(X) correlate. That is, oftentimes the map at location X[i,j] is similar to that of F(X)[i,j]. This does not hold for fully connected networks, since neurons do not contain spatial information. However, to what extent this matters is probably an open problem :-) .
Residual networks are always built with convolutional layers. I have never seen residual networks with only fully connected layers. Does it work to build a residual network with only fully connected layers?