[err4nt]

Looking at a real horse's teeth shows how worn down they are over age. Looking at a horses teeth is to check its age.

The idiom could be modernized to: 'dont check the odometer of a gift car'

[Menge]

Yes, and the assumption is that a horse is worth between $0 and some number so there is never a reason to refuse a gift horse.

In the case of other "gifts" the value can be negative. The trojan horse, software, and blankets from Lord Jeffery Amherst all need careful examination.