Does the probability of rolling a 5 or 6 on a six sided dice increase with the more dice you roll?

Of course, if you're talking about the probability of rolling "at least one 5 or 6."  An easy way to quantify it is to consider the probability of NOT rolling a 5 or 6, not a single 5 or 6 on any of the (many) dice you are rolling.  If you are rolling only 3 dice, your chance of NOT rolling any single 5 or 6 is (2/3)^3 = 8/27 = around 0.3, but if you are rolling 10 dice, your chance of NOT rolling a 5 or 6, not on any of them, declines to (2/3)^10 = about 1.73%.  Or if you're rolling 34 dice, your chance of getting no 5's and no 6's is 1 in a million.

The probability of seeing at least one 5 or 6 will change as you add more dice, but for each die, the probability is unchanged (1/3)