I have a drain on the floor of the utility room that backed up when the main drain was clogged. It was a mess, needless to say! Why does the utility room have this drain? My washer drains directly into the utility sink. And is it possible to close or remove it from the drainage system so I can have the floor leveled and tiled? My home is a 1927 Tudor.

Those floor drains are there to carry away any spillage the occurs in the basement. The down side is that it's the lowest spot in your basement and the first to back up. But there's a cure for that called "Flood Guard" (see image). These are floor drain check valves and they prevent the floor drain from backing up. Check them out at;Flood guards for floor drains - check valve to prevent flooding from FAMOUS PLUMBING SUPPLY.
I would keep the floor drain and have the tile man slope to it when he tiles. If you ever have a bad spill in your basement you'll thank me for it. If you still wish to do away with it simply fill the trap and bell with cement as you level the floor. Good luck in whatever you decide on, Tom

Even though the washing machine drains elsewhere, all things break in time. And as Speedball said that's when you'll be thankful you have the drain. I was thinking of hooking up a drain to my sump pump but I too was worried about back ups. This type of check valve is pretty cool.