Rubbing the donor area after FUE with damp tissue. Could this lead to side effects or abrasion?

It is best to ask your surgeon about post operative care as each doctor has their unique regimen.   Wiping off ointment as you describe it should be fine.

Donor area is generally dressed for first 24 hrs. After this time the dressings can come off and antibiotic appointments (Such as fucidin or polyfax) can be applied thinly and interestingly it heals very fast. Hope this helps.

abrasion after hair transplant can lead to graft displacement if performed in the first few days so do not do that