Evoked potentials can be loud, and an MRI is very loud (usually requires hearing protection). Some doctors prescribe the MRI as a standard test to rule out very uncommon things, like an accoustic neuroma. Usually doctors look for physical damage in an MRI, but if you know the likely cause for your H (for instance, lets say noise exposure), you can just tell them to put things in perspective, and maybe not take that test now, or wait a couple of months and see if your H improves. Actually you have to sign consent for an MRI test and one of the listed issues about the MRI that is stated in the document you sign is that is very loud.
Other than that, tests like audiogram, CAT scan are quiet. The LDL test can bother you a bit, but you can just tell them to stop the beeps after they go too loud for you. When you take the LDL test the beeps stop as soon as you puch a button (with some equipments) or as soon as you raise your hand. So even if it bothers you for a bit it would be just a moment.