Is it recommended (necessary) to have waterproof footwear in August? We will be on the August 18 - 31 Best of Ireland Tour.
Ireland is "green" for a reason. It rains. Not all the time, and seasons vary. You can check on-line weather sites to see the year's rainfall averages. On a tour I assume you will be inside some sort of transit for part of the day rather than hoofing it in a downpour. I take weather-resistant black (no logo) runners as my basic footwear, worn on the plane where luggage weight matters, and pack a pair of light slip-ons for the evening if I get soaked.
I didn't expect to get so wet in July - but we did and I was so glad I had waterproof footwear. (Northface Gortex). Be prepared. make sure your Jacket is also really waterproof. Mine was not (although marked as waterproof). I am going back to Ireland this year and I have invested in a Gortex Marmot waterproof jacket to wear over fleece. I have already tried it out in a downpour here. I won't make the mistake of having a poor quality rain jacket again. You will be outside a lot so make sure you are prepared.
Any time of year you go to the British Isles you'll want footwear that can stand up to sloshing through wet streets, if not boggy country fields. Doesn't mean you have to have Wellies or hiking boots, but forget about sandals. I wear regular leather fashion boots with a low heel -- the kind you can tuck your jeans into in the daytime and wear with tights and a dress for evening. Then I bring a pair of stretchy slip-on flats to use as "bedroom slippers" so I know I'll have something dry to put on in the hotel.
You'll also want a good raincoat, a rain hat, and a collapsible umbrella (though sideways "lashing" rain can make umbrellas useless).
Ireland is the green and rainy place, but not all time. You will check the online weather before you went your tour. But some time seasons vary and can’t estimate to weather so, according to me in this condition you can carry leather boots. Leather boots are waterproof and you can wear regularly also.