The water supply pipes in your house's plumbing may make noises or vibrate for a number of reasons, including water hammer or broken straps. Ignoring the vibration in your faucet's hot water supply pipe can lead to serious problems with the pipe, since the vibration could be the early warning sign of major problems.
Water hammer can occur in either hot or cold water supply pipes when you suddenly stop the flow of water through the pipe by closing the faucet's valve. You may hear a loud bang followed by the pipe vibrating for a moment. The vibrating comes from the rapidly traveling water suddenly hitting the closed valve and the collision sending powerful shock waves through the water in the pipe. A plumber can fix the problem by installing an arrestor on the water pipe, which stops the shock wave from traveling down the pipe. If not corrected, water hammer may cause pipes to rupture.
When a hot water supply sits long enough without water flowing through it, the hot water in it cools off and so does the pipe. As you run the faucet's hot water, hotter water flows into the pipe. This new water can quickly heat up the pipe, causing it to expand so quickly it vibrates. The vibrations die down after you run the faucet's hot water for several minutes, but they reappear when you use the hot water again. Wrapping the exterior of the pipe with pipe insulation helps keep the pipes warm so that they do not expand as much and, therefore, stop vibrating.
Water supply pipes need to be secured with metal straps periodically so that the pipes do not vibrate. If one or more of the straps on the hot water supply pipe that goes to your faucet has broken, then the hot water supply pipe would vibrate while the cold water supply pipe does not. If you are able to pinpoint the location of the vibration, you can cut into the wall or ceiling and see if a strap is broken or if the screw that held the strap in place on the house's frame stripped out.
The hot water supply pipes for your faucet may have a design flaw that is causing the pipes to vibrate when water flows through them. If a valve on the hot water pipe is too small, it may affect the water flow adversely. Inappropriately sized pipes also can cause them to vibrate. Have a plumber look at the pipes when they rattle, since a plumber can assess if the hot water pipes have a defective design that needs to be corrected to stop the vibrations.