Even though it doesn't discharge a "white" flash, your SU-800 does have a discharge tube that emits near-infrared to infrared light when it is fired. Just as a normal flash's tube can wear out and cease to produce light when an electrical charge is passed through it, it sounds like the tube in your SU-800 might have burned out. Short of reading through the manual of every Nikon SB flash, I couldn't find any online reference that outlined specific beep codes for the SU-800 or other flashes within the Nikon system. There's no reference to error beeps in the Manual, either. I did find some anecdotal references to six (short?) beeps as what Nikon flashes do when they refuse to fire due to being overheated.
You could send it in to see if it can be repaired, but it will probably cost more than the price for which used, or maybe even new, SU-800 controllers are currently selling.
There is one thing you might try before writing the SU-800 off. Try removing and replacing the battery in the SU-800 while the power switch is set to 'On.' It's mentioned on page 129 in the User Manual.
Quite frankly, optically controlled off-camera flashes are getting very long-in-the-tooth. Radio has become a more reliable way to control off-camera flash and avoids most of the problems encountered when using an near-infrared or even white light optical control system:
- Distance limitations. Most optical systems, especially when using a weak built-in popup flash or a near-infrared unit as the controller, are much more range limited than radio systems. Radio tends to have a greater range, especially in bright light.
- Positioning limitations. Most of the optical controllers only cover an area about as wide as a 24mm lens on a FF camera. If the remote flash, or remote receiver on the camera, is further to the right or left of this cone it may not receive any optical signal, even if it is only a few feet from the transmitter! Radios transmit in all directions from the camera or remote trigger.
- Line-of-sight requirements. In addition to being in the "cone" of light transmitted by the master (or remote when triggering a camera remotely), off camera flashes and the receivers on remote cameras must have a clear line-of-sight to the master with the optical receiver pointed in the direction of the master. This inhibits being able to place optically controlled flashes inside modifiers, placing them behind objects in the scene, placing a remote camera out of line of sight of the trigger, etc. Radio systems are not limited to line-of-sight and can even be used on the other side of walls and other obstructions (although the obstructions may reduce the range somewhat).
Difficulty with bright ambient light. Especially outside under sunlight, the power of optical wireless control is very limited. Again, especially with a relatively weak built-in popup flash or a near-infrared transmitter, the master just doesn't have much power to cut through the bright sunlight and the receivers can't detect the weak signal from the master over the very bright sunlight. Radios work just as well in bright sunlight as they do in a dark studio.
Multiple Photographers. Radio has the ability for more than one set of the same type to be used in proximity to one another without interfering with each other. (Think several press photographers all using Canon covering an event for multiple publishers. Or more than one shooter at a wedding.)
In the past few years the cost of controlling multiple flashes via radio has become much more affordable while in many cases also become much more reliable. The popularity of third party radio flash systems, including flashes with built-in radio receivers, has forced both Canon and Nikon to begin making the transition from optically controlled to radio controlled flash systems. Canon started in 2012 with the introduction of the 600EX-RT. Nikon introduced their first radio flash, the SB 5000, in 2016.
The products from Canon and Nikon aren't cheap, but there are many third party makers that are making flash units just as capable that cost a lot less. Some of the more reputable ones approach the reliability of the Canikon units while others are priced more as 'disposable' that allows one to always have spares on hand. There's also nothing keeping a photographer from attaching low-priced radio transmitters/receivers to their camera/existing Nikon flashes. The extra pieces and battery management required is a bit more of a hassle than using flashes with built-in radios, but it is a solution many find best fits their needs.