A DVD player will work fine, as both the RoI and the UK use 625/PAL system I.
TV's intended for the British market will also work in the Irish Republic, but may not be able to receive all broadcasts. British TV now uses UHF channels (21 through 69) exclusively, and after the old 405/VHF service was phased out, TVs came fitted with only UHF tuners.
In the R.o.I., however, they still use VHF channels in some regions of the country, so you would need a TV with a VHF tuner. The Irish VHF channels are labeled A through J (although channels A, B, and C are now no longer in use). There are many transposer (relay) stations, but in general if you're receiving from the main stations. you could get by with UHF around Dublin, across much of the midland area, and in the northeast. You will need VHF for most of the south and west.
With many modern TV sets now being sold for multiple markets, it's not uncommon to find U.K. sets with VHF tuners once again, but it depends upon the make and model. Obviously, if you want to record programs off air you would need a VCR with VHF inputs as well in some areas. These are much less common in the U.K. market. A DVD recorder would obviously need VHF facilities as well if you want to record off-air.
Hope this helps.