Can I give my follower potions?

In Diablo II, you could feed a follower potions by dropping them from your inventory onto their portrait. It wasn't very practical in the heat of battle, but it was possible.

I've tried this in Diablo III, but it didn't seem to do anything. Can I no longer use potions on followers, or is there a different mechanic for doing so?

You cannot give them potions, but they do receive healing from health globes that you activate. So the best way of healing them (bar healing skills) is to jump on these globes.

In general, if you find them "dying" too much, you can also consider changing their gear to be more defensive in nature, and to change their skills to the more defensive ones.

As @Oak pointed out, they do get healing from the health globes (and I assume share any bonuses for health pickup).

Also remember they get 2.5x benefit from any stat boosting equipment, so giving them equip with Vit will help a lot.