Will we need to pay for electricity and water when we rent a vacation home in Costa Rica?

Not if you are renting it short term...

Utility is included in the daily or weekly rental rates. However, some budget accommodations offer you an option to have air conditioning at an extra cost per day or week.

Also, if you are renting a vacation home for longer than a month, you may be responsible for utility charges, if so, it should be specified in your rental contract.