I raise to $10 with two black 10's at 1$/$2 blinds from middle position with $150 behind. Villain has $200. Everyone else folds

Flop is Q64sxx I check he bets $25 I call turn is Js checks around. River is 2s. I check he bets $65. I tank for about a minute. Story was
Remaking sense. Why would he bet a Q the check a J when draws pop up on the turn. Really narrowed it down to him back sorting some weird flush or nothing. I called with those 10's and he turned over 7's like they were the nuts.

So what's the right line here? Am I setting myself up for nastyness?

It's really hard to offer advice when there's a lot of key info missing. (Probably why no one's offering any help.)

What kind of player is the Villain? What's you're image? How often do you raise? How often does he call? What position is v in? What range do you put him on?

The way the action plays, you're playing too timid. If you had 3 or 4 players calling your (too) small pre-flop raise, then I could see you check/folding to a bet. But heads up, YOU should be betting the flop. If v calls, then you can check the turn for pot control or to induce a bet. If he checks back, you can be pretty sure your tens are good and bet the river for value.

You probably got the most money from v because you played your hand so bad. (It looks like you've got AK.) But playing like this is making things harder on you... and will make you very exploitable.

Stack sizes would help and some sort of description of villain really matters here. Against tight villains river is an easy fold as he will have a lot of weakish QX and some weak JX in his range. Against a really bluffy one river is almost always a call.

Against most this is a line you can use occasionally in this situation where you have a weak made hand and the board is fairly dry. It is very easy to exploit if you use it very often though. It makes you too passive in game and a decent villain will be able put you on that sort of middle pair and only bet river if they are beating you or bluffing.

It would help if we had more info on V. I agree with you that this line by V doesn't make too much sense, I would probably sigh call here too.