Doors and mounts

The best scenario is to get an unlocking item. Which is either a credit card, a lockpick, or best of all a skeleton key. You can apply these items while still mounted in order to open the door efficiently. And, as an added benefit, keep the door in operation should you need it for anything else. A wand of opening or the spell of knock will also have the same effect, but the tools are free to use.

Failing that, if you have a wand of striking or the spell of force bolt, you can always break down doors with the ray and maintain your mounted position. Be prepared for any consequences like shattering items or hitting creatures that are behind the door, though.

If you lack these kinds of items, you'll need to either find another way into the room (perhaps a wand of digging, if you want to just make your own shortcut), or you'll have to hop off and give the door a swift kick yourself before mounting again.