I was trying to figure out whether a particular TV remote was working at all, or just had the wrong code, when I remembered that cellphone cameras can see infrared light. If you turn on your cellphone camera, and use it to look at the remote (while pressing a button on the remote), it should show a light from the LED as shown in the picture. If there is no light, check the batteries, clean the battery terminals, and try again. If still no light, the remote has probably failed. Nifty!
This also works to see infrared light from security cameras.