General:
* Ignore
* Disregard
* Overlook
* Disregard
* Skip
* Pass over
* Dismiss
* Forget
* Fail to attend to
* Take no notice of
More specific:
* Abandon (to leave someone or something completely)
* Derelict (to leave something in a state of disrepair)
* Neglectful (adjective form, emphasizing the act of neglecting)
* Inattentive (lacking attention)
* Careless (lacking care or attention)
* Unmindful (not paying attention to something)
* Remiss (failing in one's duty)
Informal:
* Blow off
* Brush off
* Shun
The best synonym will depend on the specific context and the nuance you want to convey.