Will cleaning the quote tables interfere with abandoned cart emails?

Yes, this can interfere with abandoned cart emails, because the abandoned cart email operates directly off of the quote table.

What you want to do is just  make sure that you are not cleaning up quotes that are at least as old as your furthest out abandoned cart email.

So for example, if you have three abandoned cart emails - one at first hour, another at day 2, and another at day 4, then you should be safe cleaning up quotes that are 5 days old or more.

Also keep in mind that if you intend to use the abandoned cart blast email, that can send out to abandoned carts that are quite old - anywhere up to 200 days or 300 days or even further depending on how you want to set it up.