First, you may find it helpful to read about exhibit states.
The short answer is no, error retries and resyncs do not count against your quota. They are "free".
The long answer
Once every 24 hours, OrbitKit moves 20 exhibits for each platform from WAITING to PUBLISHING. Once an exhibit is in PUBLISHING state, OrbitKit tries its hardest to publish that exhibit on the venue -- retrying approximately every 8 hours for about a week. This doesn't slow down the queueing process, so yes it's possible for exhibits to "pile up" if something is seriously wrong.
There are a lot of things that can go wrong. Some are the fault of the platform (transient errors that simply fix themselves on retry); some are 'your' fault (a common one is deleting the blueprint!); some are 'our' fault (a new edge case, bug, or a change to the underlying platform that we need to adapt to).
Some errors (like a deleted blueprint or hitting an platform upload limit that requires your interaction) will move the exhibit to BROKEN state immediately. After you fix what is wrong, you can put these back into WAITING state with a button on the dashboard.
Resyncs are free in the system and don't count against your quota. However, they are queued like everything else and thousands may take significant time to complete. A published exhibit that is resyncing will be in UPDATING state.
If an UPDATING exhibit fails too many times, it may go to BROKEN. If you "unbreak" this exhibit, it will go back to UPDATING and will not count against your 20.