What are some reasons that business owners choose to prevent cyclists from using the drive-thru? Are there legal issues?
Having cars in a drive-through is easy because they can only go by the ordering|payment|delivery windows one at a time.
Any bicycles or pedestrians in that funnel could overtake each other and make a big confusion out of the sequence. That would ruin the process for the business because it's specifically meant to only handle an unchanging sequence of customers.
Plus, drive-though is faster than going in. If lots of pedestrians walk through the drive-through then you'll end up having a long queue inside and outside which also ruins the business model for the drive-through.
When I tried it at McDonalds, they told me they were only insured for motorized vehicles coming through the drive through and would not serve me.
They had a campaign on at the time promoting something along the lines of how their food was healthy when enjoyed as part of a healthy life-style. I pointed out that I was cycling as part of my healthy lifestyle, and they really ought to do everything they could to support me with that. When I emailed them later to ask more about it and suggested they allow bikes through the drive through, they blatantly ignored my suggestion and replied restating that their insurance didn't cover them for bikes.
When I mentioned to the person at the window that I didn't have my bike lock, they told me I could take my bike inside! I wish now that I'd taken them up on the offer - I'm sure I would have struggled through their double set of doors, and my bike would have become an obstacle to other diners - but I was a bit annoyed at the time so I went elsewhere to eat. :)