Sure!

= good

= bad
7.62 round in 7.62 chamber

.308 round in 7.62 chamber

.308 round in .308 chamber

7.62 round in .308 chamber

- Again, your #### probably won't fall off, but when dealing with firearms, it's better to be safe than ####-less!
AFAIK, the 5.56 vs .223 and 7.62 vs .308 debates are essentially identical. Shooting a 7.62 cartridge in a .308 chamber should not cause a catastrophic failure; however, due to the slightly wider throat specifications in NATO chambers, firing either 7.62 or 5.56 NATO ammunition in .308 or .223 spec chambers will result in greater wear on the barrel throat and will reduce its service life.
AR180 Shooter;
A quick thing: the 5.56/.223 and 7.62/.308 debates are NOT the same though to many they might be. FYI I used to engineer both of these weapons' systems (C1A1, C2A1, C3 & A1, as well as C7, C8, A1s, LSWs, C12, C13, etc) for DND and through contacts in DCRA did a lot of work on compatibility. SO...
5.56/.223 is as mentioned a throat configuration issue and with the stupidity of the US DOS we have all kinds of guns with markings that mean nothing compared to the chamber. The only safe way to deal with these things is to A. call the manufacturer (most of the serious companies build them to handle both - Colt in AR15A2s and more recent, Steyr AUG (only Scout and SBS are .223)) or B, get a chamber cast by a gunsmith and examine the shoulder diameters, leade, and shoulder to throat angles. So the previous smile gram is generally accurate BUT DOS has screwed things the last 2 years and I know for certain S&W refuses to answer the question. You can certainly get issues with 5.56 in a .223 chamber and if you are shooting tracer (Belgian L110, Canadian C78, US M856) you sure can have an issue as the bullet goes way back into the case causing the pressures to increase (hence the geometry change for NATO). Hand loading for bolt guns where you can load the bullet out so it doesn't go into the powder space is a much better solution though approach with care per any hand loading venture. For semis where mag limits are a concern requires proportionally more care in load development.
On 7.62/.308 it is NOT the neck and leade that is an issue. (In .223 we are going from light 55gr to heavy 62+gr bullets in the debate in .308 the MilSpec bullet is quite light at 147gr nominal.) The problem can be at the base in the case (this can also be an issue if you load 5.56 velocity and bullets in commercial brass). If I get all crazy some night I will take pictures of some bullets in case mouths and sectioned bases in .308 to show this for sure (but I digress).
The MILITARY case is required to be much thicker through the base than the commercial equivalent. Ever wonder why those loading manuals say back off 2 grains? Well now you know, there is that much extra brass in a military case compared to a civilian case. (Correctly pointed out as due to the full auto guns tendency to really rip on the case to get it out of the chamber and in the case of the G3, while it is still pretty highly pressurized. There are also cost implications if you can save that brass on several million cases you make more profit! And other things but we again digress.)
So what right? Well you get into last point of support on the base of the CASE issues with civi ammo in military guns. Not all of them BUT the FAL design is famous for leaving a lot of case out of the back of the chamber when it is in battery. Look at your Military brass after firing in your FAL versus say your Remington 700 (factory!) and see where the brass swells to: quite a bit further forward in the military piece. Commercial brass - Remington is the worse I have seen regularly for thinning the case base but others are close - can often let go at the base in these less than fully supported Mil Spec barrel/chamber designs. In normal Mauser, Remington, Winchester type bolt rifles with factory barrels you should never even notice a hitch with either kind of ammo.
Lastly in the .308/7.62 debate be REALLY CAREFUL when you buy a rebarreled gun or rebarrel a gun. I had several pictures and ruptured cases sent to me in the 80s that had a chamfer on the back of the chamber made to ease loading for single shot target rifles. With military brass it held together. With some foreign and civi brass it had a habit of occasionally blowing out a case base and sending gas into the breech. If in doubt and you are loading a new non-NATO Spec ammo into your FAL, M14, HK91, etc. pull a bullet and section the base. Do a few from Remington, IVI, Winchester (commercial) and Lake City and check it out for yourself. With commercial guns caveat emptor and do some checking or ask your gunsmith to look into it.
Safe shooting. Of anyone has specific gun combo question message me privately please.
Phil