There are two types of remote controls first one is classic IR remote controls and second ones are newer bluetooth IR remote controls. Both work by transmitting the infrared signal to TV or the devicce they are control. 

How to check if a Remote Control is working

Test with New batteries

Before you start testing your remote control remove the existing batteries and replace with new batteries, this will ensure your not wasting time just because of drained batteries.

Free Stuck buttons

Now that we you have ruled out batteries, next step is to find if there are any stuck buttons. press each button couple times to see if you can free any stuck button. 

Check if Remote is sending Infrared signal to TV

To check if your remote is sending IR codes , you need a Digital camera or a phone with out IR filter. Some phone and couple models of iPhone has a built-in IR filter which does not show the infrared light. check if you can disable the IR filter on the camera in settings.

Step 1: first Turn on your digital camera or camcorder or the camera on your mobile phone.
Step 2: Point the end of the remote control with a the IR emitter to the viewfinder or camera on phone.
Step 3: Press and hold one of the remote control buttons. press green power button for newer Bluetooth remote controls.
Step 4: Look at the viewfinder or screen of your camera or mobile phone.
Step 5: If the remote control sends a signal, you should see a light in the camara screen.

When to change the remote

If you see a continuous light from IR emitter or you don't see a light when you press button on remote its time to replace the remote control with a similar model or a Universal remote control.