Recently when running a load in the dryer, we noticed the front of the dryer, under the door, was getting hot to the touch. Is that normal for a dryer to be hot to the touch in front when the dryer is running?

i assume you mean the lower part of the front panel gets hot to the touch. i'd say that's NOT normal. you've checked the obvious things, which is good. i believe there are vents, of some kind, along the front bottom of the machine (where the machine draws IN air, according to one site). check that those are clear.

after that, my thought is that the drum is somehow venting to the interior of the machine. other parts of the casing are NOT hot to the touch? i don't know why/how it would vent only to the lower front, and/or only heat up the lower front of the casing. you don't hear grinding noises? (that might indicate a problem with the drum support glides.) no other strange noises? (something to indicate a problem with the blower?)

i'd try taking off the front panel, and checking: 1) that there's not some strange lint build-up below the lint trap; 2) the blower--if that's overheating, maybe it's heating the front of the casing; and 3) the felt trap duct (it should keep heat from escaping the drum).

for the last one, there's a video here:
GE WE09X20441 Felt Trap Duct - AppliancePartsPros.com

you can look at parts schematics here:
Parts for GE DCVH680EJ1WW: Cabinet Parts - AppliancePartsPros.com

and if you click on a part name in the list below, you'll get some extra info on each.

i found these pages/sites on GE dryers, but they weren't much help:
General Electric Dryer Repairs | GE Dryer Repair Manual
http://products.geapplia...-content?contentId=18709

and another site about repairs (repair c l i n i c with a commercial web address, and secured as an https) and if you append this, it's about GE dryers. but it didn't have too much relevant info either.
/RepairHelp/How-To-Fix-A-Dryer/6-3-849747-/GE-Dryer-overheating-DCVH680EJ1WW