An informal expression meaning that something (e.g., a phone or alarm) keeps ringing continuously; '-っぱなし' indicates a continuing state.
深夜に隣の家の電話が鳴りっぱなしで眠れなかった。
I couldn't sleep because the neighbor's phone kept ringing all night.
目覚ましが鳴りっぱなしだから止めてくれない？
The alarm keeps ringing — could you turn it off?
外の工事の音が鳴りっぱなしで集中できない。
The construction noise outside keeps going, so I can't concentrate.