Since most mounts do not even have an escape artist skill, I feel like this feat and a good portion of mounted combat in dungeons would be useless. Am I missing something?
The feat Tunnel Riding (RS 145) lets you and your mount ignore penalties AC and attack penalties for squeezing and to fight while squeezing.
In some cases, you may have to squeeze into or through an area that isn’t as wide as the space you take up. You can squeeze through or into a space that is at least half as wide as your normal space. Each move into or through a narrow space counts as if it were 2 squares, and while squeezed in a narrow space you take a -4 penalty on attack rolls and a -4 penalty to AC.
When a Large creature (which normally takes up four squares) squeezes into a space that’s one square wide, the creature’s miniature figure occupies two squares, centered on the line between the two squares. For a bigger creature, center the creature likewise in the area it squeezes into.
A creature can squeeze past an opponent while moving but it can’t end its movement in an occupied square.
What the Tunnel Riding feat doesn't do is let you do is ignore the next part of squeezing:
To squeeze through or into a space less than half your space’s width, you must use the Escape Artist skill. You can’t attack while using Escape Artist to squeeze through or into a narrow space, you take a -4 penalty to AC, and you lose any Dexterity bonus to AC.
Emphasis mine. So, yeah, riding an elephant through the kobold warrens is challenging, but riding a horse or other Large mount in a dungeon is slow and clumsy without the feat Tunnel Riding, and with the feat riding that Large mount around the dungeon becomes a fairly viable tactic--assuming, of course, that the DM's dungeons use standard 5 ft. squares.
To Answer the Title
Yes, a mount must still make an Escape Artist check to fit into very small spaces if the rider has the feat Tunnel Riding, but a Large creature traversing a hallway at least 5 ft. wide doesn't.