In this hand I pretty much had put villian on a pocket pair as he had been very nitty raising pre-flop up to this point. Clearly this wasn't the case, and the min-raise didn't fit into this mould. Looking back on the hand a range just including pocket pairs doesn't seem to fit.
Called the bet perhaps mostly on a whim, I know I will likely have to fold my hand a large amount of the time on the flop figured it was worth a call for the price offered.
As played, on turn would best play have been to flat call, or was it ok to try to get extra money into the pot while we assume we are ahead of villians range?
With flush draw, full-house draw and made trips I thought I was likely favorite or at least flipping against a range?
Known players:
[tg]Position_Stack:hands/vp$ip/RaisePreflop/AF/WTS/ORL/BBvsSteal/W$SD;Hero$2.09:///////;MP1$4.33:147/16/14/4.8/33/0/50/75;[/tg]
0.01/0.02 No-Limit Hold'em (9 handed)
Hand recorder used for this poker hand: PokerStrategy.com Elephant 0.102 by www.pokerstrategy.com.
Preflop: Hero is BB with T♥, 4♥
2 folds, MP1 raises to $0.04, 5 folds, Hero calls $0.02.
Flop: ($0.09) 5♥, 4♠, 4♣ (2 players)
Hero checks, MP1 bets $0.09, Hero calls $0.09.
Turn: ($0.27) 8♥ (2 players)
Hero checks, MP1 bets $0.16, Hero raises to $0.32, MP1 raises to $0.61, Hero raises to $1.26, MP1 raises to $4.05, Hero calls $0.70 (All-In), MP1 gets uncalled bet back (All-In).
River: ($4.19) J♣ (2 players)
Final Pot: $4.19
Results follow (highlight to see):
MP1 shows a straight, eight high (7h 6h)
Hero shows (Th 4h)
MP1 wins with a straight, eight high (7h 6h)
You can't hit much with the hand. 
If he has a hand then he is going to pay us anyways if not then wont pay much either. But as played just 4bet-shove yourself and I doubt that I am getting away from here. He may sometimes even overplay any kind of overpairs.