An excellent question and it shows that you are thinking.
Everybody knows that a bolt gun has a stronger action than a Mini-14 or an AR-15 so it must be ok to shoot 5.56 NATO in a SAAMI .223 Remington chambered rifle. Or is it.
The .223 Remington has been around a long time and there are many bolt action rifles that were chambered for the cartridge in the 60s and 70s as varmint rifles. These rifles gave excellent accuracy because their chambers were tight and the leade was short. Every body used light weight short bullets for varminting. As time went on and the AR-15 moved up in bullet weight in the 1980s (62gr) and then got adopted wholesale for competition, with even longer bullets, the chambers grew in length and some bolt action rifles (certainly more recently) have tried to keep up.
However, I would say the vast majority of .223 chambered bolt action rifles are SAAMI .223 chambered and may be subject to the additional pressure engendered by the bullet of a 5.56 NATO cartridge being jammed in the lands. Some bolt action will not be bothered by that extra pressure, especially the newer rifles, but the older ones may well be stressed.
One thing to consider is that most bolt .223 have a barrel twist of 1:12, thus restricting the bullet weight to 55gr or less, thus not NATO caliber. Also, bolt action rifles do not consume as much ammo as semis such as the AR or the Mini, so the need for cheap surplus ammo is not as strong.
So if you have a bolt action rifle with a 1:12 twist, why would you want to shoot 5.56 NATO wth its longer 62gr steel penetrator bullet? You will probably not stabilize it and why pay with extra pressures?
But let's say that you really want to shoot 5.56 NATO in you bolt rifle with the 1:12 twist. Here is a quick test to help you make your decision. Take a 5.56 NATO bullet and with a marker, paint the ogive of the bullet black, heck paint the whole exposed bullet. Then in a safe manner, chamber the round, if you can. If it chamber without you pounding on the bolt, open the bolt and slowly eject the unfired cartridge. Now examine the bullet for any scuff marks to the paint. If you see none, the chambered round did not have the bullet in the lands. If you see scuff marks, the bullet was touching the lands are you are now aware and warned about the situation.
People have been shooting surplus ammo, including 5.56 NATO ammo in all manners of bolt guns for a long time. I have never heard of any failures with that combination, but that only means that I have nt heard about any; it does not mean it has not occured somewhere.
Personally, I specify a longer leade for my .223 rifles, but with a tighter chamber than 5.56 NATO and even .223 Remington match in the case of my match rifle. I have zero interest in varminting or using light short bullets. And I would never use surplus ammo in my rifles; there is a reason it's considered surplus.
Does that answer your question?