I might use "What's your day like?" somewhere around the middle of the day, when the listener already has a good idea of how the day is going. The answer might be "It's OK, thanks. Pretty busy but I can handle it. I'm going out for dinner later though".
I might use "What's your day look like?" (in which "What's" is a contraction of "What does") at the beginning of the day when I want to know what sort of things a person has planned. The other person might say "Hang on, let me check my calendar. Hmm, it's looking busy. I've got meetings at 9.30 and 11.30, then lunch with my boss, then staff appraisals all afternoon. I won't get out of here till about 8pm!"