Specific tiling for washing machine closet?
I agree with ecnerwal rubber pads would be best. We are a diy site and here to help you. 
Other things

I prefer to use a concrete backer board screwed down to the flooring to mount tile to. Never put tile directly on chip board , OSB or most underlayments. Plywood will work but the others when wet come apart as the tile can allow moisture and then it becomes a real mess.

Pick a tile you like because it can last your lifetime and I think most new to tile do better with larger sizes. I have seen many first timers try 2x2 because they are on a sheet and it usually looks diy. Larger tiles and spend the few bucks for spacers and use them. 

Watch some on line videos so you understand how to “set” the tile using a notched trowel. 

Starting with a small space like a laundry closet is a great way to get experience , remove existing mop or trim boards. Put your backer down and go for it, good luck!