I have a Tier 2 General UK visa which is valid until 2019. Can I travel to Ireland (Dublin, Belfast) without an Irish visa? It would be for a short holiday.
You need a visa to enter the Republic of Ireland unless you are in a visa exempt nationality.
As a general rule UK visas and leave to enter or remain in the UK are not valid for Ireland as it's a different country. Certain holders of UK visas qualify for a visa waiver but that's limited to certain nationalities, and according to the Irish immigration authorities only applies to short-stay UK visas.
You could take a trip to Northern Ireland on you visa as this does not involve leaving the UK.