Maps will cease to indicate current position of the player holding it. Maps placed in frames will continue to display the green arrow of the frame they were placed after removing them and running around, either open or not.

Conditions tested:
Fresh map displays properly when opened, player moving.
First placed in item frame, displays green arrow within map properly.
Taken out, displays player indicator and green arrow of frame.
Replaced, displays green arrow properly.
Removed a second time, same as before. Once expanded, no longer displays green arrow (displays properly).
Expanded map placed in item frame, displays properly.
Taken out, green arrow remains, overlaps player indicator.
Map expanded a second time, displays properly, even after placing in item frame and removing. However, placing in item frame, logging off, and rejoining server, map retains green arrow after removing from frame.
Copy of map behaves properly in regards to filling in on item frame while exploring with original; however, both retain green arrows when held, even when both copies are in hand, and none remain in item frames.
Map expanded a third time, initially displays properly; green arrow remains after logging off and rejoining server with map in frame. Otherwise, maps behave as normal.
Map expanded a final time, behavior remains consistent as before. Map displays properly when exploring except for retaining green arrow.

...

So I couldn't figure out how to replicate the bug. I resorted to taking screenshots of the faulty map to prove I wasn't going mad. I placed the fully expanded test map in the adjacent item frame to my stationary bugged map (fully explored and expanded) and placed the respective copies in my hotbar, then took a screenshot of me holding the bugged map and looking at the item frames, for initial comparison. Now, moving away from the item frames, the previously fine new map, which I could not replicate the bug with, has replicated the bug. So, I expanded one of the other test maps, since they do work as normal fresh from the crafting table, and returned to where I took the two experimental screenshots to get a control image. I then went back, placed this new, fresh map in the item frame, took it out, confirmed there was no lingering green arrow, replaced it in item frame, logged out, logged back in, removed it, and the green arrow had returned with all the steely resolve of Oliver Queen. This is the fifth screenshot.

The server is SMP, non PVP, 1.8 .exe straight from the website.