I know that the cards I'm familiar with such as Act of Treason and Act of Aggression all specify gaining control until end of turn. I'm wondering if something such as Illusory Gains, Beguiler of Wills, or Blatant Thievery would take control of a creature permanently. I know that the answers on this question Does Illusory Gains make someone else's Commander my own forever? indicate that is not the case but I'm wondering if there is a rule that indicates that effect doesn't last when it's not specified and furthermore are there any cards that do specify the effect lasting permanently?
Control-changing effects aren't really special cases. What you're seeing here is just that some effects are permanent and some aren't. If you put Goblin War Paint on a creature, it gives that creature +2/+2 and haste - but if the aura goes away, so does that effect. It doesn't give +2/+2 and haste forever, just as long as it's around.
Specifically, continuous effects generated by an object last only as long as that object generating them is around:
611.3b The effect applies at all times that the permanent generating it is on the battlefield or the object generating it is in the appropriate zone.
and continuous effects generated by spells or abilities last as long as they say they're going to last, and if no duration is specified they last until the end of the game:
611.2a A continuous effect generated by the resolution of a spell or ability lasts as long as stated by the spell or ability creating it (such as “until end of turn”). If no duration is stated, it lasts until the end of the game.
Also perhaps relevant is:
611.3a A continuous effect generated by a static ability isn’t “locked in”; it applies at any given moment to whatever its text indicates.
So an Aura that has an effect on the creature it's enchanting always applies just to that creature, even if it gets moved around.
Illusory Gains is an example of an effect generated by an object that's not around forever in the realm of gaining control, but a more obvious one is the classic Control Magic. Its effect lasts only as long as the aura is on the battlefield enchanting a creature. If you Control Magic an opponent's creature, and they destroy the Control Magic, they get their creature back. If you move it to a second with an Aura Finesse, you'll lose control of the first creature and gain control of the second creature.
Illusory Gains is the same idea. It gets moved around, and it only says you control the creature it's enchanting, not any other ones, so when it moves away from a creature, there's no longer an effect giving you control of that creature.
Another way a continuous effect might not last forever is if it's created by a spell or ability which specifies a duration explicitly (like "until end of turn"). This is what's going on with Act of Treason. That case is pretty obvious from the text on the card.
The last case is if the spell or ability which creates it doesn't specify a duration. In that case, it lasts until the end of the game. So Beguiler of Wills and Blatant Thievery give you control of things indefinitely: they just say gain control, and don't say "...until end of turn" or anything like it to specify a duration.Since the rules work this way, there aren't cards that explicitly say "gain control forever"; they don't need to.
Just do a search for "gain control" excluding turn or "end of turn". Many of those cards have conditional control, but they last past the end of the turn.