It's awful hard to say with just the limited info but, if it is consistent with the vertical stringing only, common things, in order that I would check. It's actually not as bad as it sounds if it's just vertical. Usually it can be found and fixed.
1. The stock. I'm guessing it's the stock and would work on that 1st. Factory Remington 700's have a bump in the fore-end area to provide upward pressure on the barrel. Yours may be getting warmed up/hot as you shoot it and more upward pressure on the barrel as it gets hot.
I had a factory 30-06 that would make one continuous vertical string when shooting 5 in a row from sandbags. Check to see if yours has that pressure point. It may just be the fact as it gets hot, it will string vertically.
If factory, you may want to see if aftermarket stock that's properly bedded will help. I seldom keep a factory stock on any target or varmint rifle unless it's been bedded. Mine immediately stopped the vertical stringing and stared putting them in a 1/2" hole after nothing more than swapping out the factory stock.
2. Check all the stock hardware to see if it is properly tightened down.
3. Check a different brand ammo or try several different loads. It can be tricky as different primers, powders, and bullet brand/weight will make a difference. Reading my Hornady mannuel states IMR 4198, VIHT 130, 135, and H 322 gave best results. I like Vihtavuori 130. It meters well and is very consistent. Try several weight bullets, I like Hornady, Sierra, or Bergers Try 40, 50, 52 match, and 53 grain match. You may be surprised how well it can tighten things up.
4. Consistent primer ignition. Make sure your firing pin is consistent, not worn out, loose, or has a weak spring. Look at your spent cases and see what kind of strike you're getting.
5. Make sure to properly torque down all scope hardware. If it's loose, tightening will change P.O.I. I don't think it's this because it will normally be all over the place, not vertical only, if something is loose with the scope or it's hardware but it's always worth making sure. You could swap out your scope if all else hasn't worked.
Good luck and please let us know what you find out!