Statistically, a top-end .308 will probably have a SLIGHT edge in accuracy over a top-end .30-06.
Assume a tuned-up 700 action, pillar bedded into the best stock, using the best Krieger barrels and carefully crafted handloads built around that particular rifle's needs, and a .308 should theoretically be capable of shooting into the .1s, certainly the .2s. A comparable .30-06 should be capable of shooting into the .2s, certainly the .3s. At least, this is what knowledgable people say are the statistical possibilities. Obviously, individual rifles can vary up or down.
Let's assume this to be true. Say you built an '06 match rifle that shot 1/4 m.o.a, or even 1/3 m.o.a. Would you be disappointed that it didn't do better and dream for more accuracy? If so, you need to stick with the .308, or something like the 6-BR.
As others have said, the sheer practicality of the .308 generally favours its use. It burns less powder, has a longer barrel life than an '06, less recoil (the biggest factor, by far), and has the largest selection of match components, chamber reamers, brass, etc.
I'll tell you one thing, though. If you didn't mind more recoil and expense, then a .30-06 would make a formiddable precision/sniper rifle, with today's components. Put together a top-end barrel like a McLennan or Krieger .3075, 1-10" twist, in a top-grade stock, load it with Lapua .30-06 brass (readily available from Lapua suppliers), Fed 210M primers, 175-grain Sierra MKs pushed by -- and here's the secret -- 62+ grains of Vihtavuori N560, and you'll see that the .300 Winchester Magnum has one less reason to exist. In addition, you'll be within DCRA rules, and be able to use that rifle on all DCRA ranges (where a .300 Win Mag is banned).
I'd also expect such a combination to be capable of spectacular accuracy.
So, no, a .30-06 is not obsolete. With today's components, its better than ever. But to really make it worthwhile, you'll have to burn a lot of specialized powder and accept considerably more recoil and muzzle blast.