Well, avast! doesn't scan the files for no reason - something must be touching them to trigger the on-access scanning (e.g. the file manager extracting the icons from the files, or some kind of indexing service - which might be in some cases hard to distinguish from real execution of the file). However, that behavior may be different on a network drive - maybe the files are not touched, or they are loaded differently, not triggering the (unnecessary, in this case) scan.
So the question is - is it detected if you really execute the file (should be safe if it's a FP?), not just browse the folder?
If not, then I'd ask - what kind of network share is it? Does your user account have rights for those files, or do you have to supply an additional password? Is the file detected if you scan it manually (e.g. from the Explorer context menu)?