Usually, no, you don't have to be subject to emissions testing. The idea is that you don't drive the car that often to begin with, so the environmental impact is very minimal because the wear and tear on the vehicle itself minimal. Meaning of course that the engine isn't expected to break apart so quickly.
However, there are some trade-offs that you should realize before opting for any sort of "Classic Car Plates". And I keep that in quotes because the types of plates/categories available differ from state to state. Not counting Horseless Carriages, California has a single catch-all Classic Plate whereas Nevada for example has 4. Some other states like Texas are the same way with multiple plates too. But more importantly, again, are the different restrictions that come with classic plates.
First off is the driving restriction. California for example specifically states the restrictions are related to historical and leisure purposes (Special Recognition License Plate Application). Nevada by comparison has a mileage limitation of 5,000 per year, but they don't care why you're driving the car. Which is something to keep in mind, but we'll cover that in a second...
States will vary of course, so you'll need to check your local regulations accordingly. County location dictates as much about renewal fees and emissions exemptions as the state itself does. So keep that in mind too.
In order to obtain Classic Car license plates so that you can get that emissions exemption in the first place, your car usually has to pass a smog test the first time before future exemptions can be granted. So be aware that the purpose here isn't to allow a polluting car a way around the testing. It is simply meant to make things easier for existing, old vehicles that are in compliance and don't see much use anyways. So the lack of a smog check keeps them off the road even more. Oh, and you also have to have met the limited mileage requirement the previous year in order to qualify in some cases too.
Likewise when you sign a classic car registration you're required under perjury to state the current mileage of the vehicle each year, and/or an oath to only use it for the purposes specified. Nevada is pretty lax stating that if you exceed mileage you just need to get the car smogged, though you might just have to surrender the plates. In some cases too, Nevada may even send your vehicle a summons to appear at one of the state's two Smog Laboratories for emissions testing if they have a valid reason to suspect your car isn't within compliance. Like if your engine is smoking, and someone reports you. I've no idea what California does in these cases. But it IS key to keep in mind that you do have these restrictions in place, potential penalties for violating them, and extra paperwork to fill out each year.
Which is in itself the big question: Is it really worth it all? Save for a physically damaged catalytic converter, if your car is properly tuned you're going to be able to pass Smog anyways. So other than for personal vanity for classic plates, what's the point for a DeLorean? Personally I like just getting a renewal notice in the mail, getting my emissions testing completed at a station, and then just either going online or visiting a kiosk to complete the renewal to be done with it. No printing up forms, filling them out, then going to the DMV in person to obtain permission for the exemption the first year. Then printing up & filling out forms/affidavits, and mailing them off each subsequent year or visiting the DMV again.