If a permanent does an effect before it is sacrificed (ie a land tapped for mana or a creature tapped for a special ability), does this effect is canceled by the sacrifice?
No. Sacrifice doesn't care about the state of a permanent being sacrificed. Sacrifice cannot cancel anything, it can only move a permanent you control from the battlefield to its owner's graveyard.
Land being tapped for mana is a mana ability, as is Orcish Lumberjack's activated ability. These mana abilities don't go on the stack, and resolve immediately so it cannot be countered or cancelled in any way (with one exception noted below).
605.1a An activated ability is a mana ability if it meets three criteria: it doesn’t have a target, it could put mana into a player’s mana pool when it resolves, and it’s not a loyalty ability. (See rule 606, “Loyalty Abilities.”)
605.3b An activated mana ability doesn’t go on the stack, so it can’t be targeted, countered, or otherwise responded to. Rather, it resolves immediately after it is activated. (See rule 405.6c.)
A creature tapped (or other cost) for an activated ability that isn't a mana ability, gets placed on the stack. It can be responded to (countered, replaced, etc.), but the ability exist independently of its source (the creature). Sacrificing the creature that generated the activated ability will have no effect on the ability.
112.7a Once activated or triggered, an ability exists on the stack independently of its source. Destruction or removal of the source after that time won’t affect the ability. Note that some abilities cause a source to do something (for example, “Prodigal Pyromancer deals 1 damage to target creature or player”) rather than the ability doing anything directly. In these cases, any activated or triggered ability that references information about the source because the effect needs to be divided checks that information when the ability is put onto the stack. Otherwise, it will check that information when it resolves. In both instances, if the source is no longer in the zone it’s expected to be in at that time, its last known information is used. The source can still perform the action even though it no longer exists.
Ultimately, is a sacrifice equivalent to bury or destroy, or does sacrifice have any special side effects on the sacrificed permanent and its effects?
Neither. Sacrifice is certainly not the same as destroy, since you cannot sacrifice a permanent that you don't control, but you can destroy permanents that you don't control. You can replace destruction with regeneration, but you cannot replace sacrifice with regeneration. Bury is an old term that is currently obsolete, and some but not all cards have received errata so that they now say "sacrifice [a permanent]". This isn't an equivalence relationship, because not all "bury" cards are "sacrifice" cards. Other cards have received errata so that they now say "destroy [a permanent], it cannot be regenerated."
701.14a To sacrifice a permanent, its controller moves it from the battlefield directly to its owner’s graveyard. A player can’t sacrifice something that isn’t a permanent, or something that’s a permanent he or she doesn’t control. Sacrificing a permanent doesn’t destroy it, so regeneration or other effects that replace destruction can’t affect this action.
701.6a To destroy a permanent, move it from the battlefield to its owner’s graveyard.
701.6b The only ways a permanent can be destroyed are as a result of an effect that uses the word “destroy” or as a result of the state-based actions that check for lethal damage (see rule 704.5g) or damage from a source with deathtouch (see rule 704.5h). If a permanent is put into its owner’s graveyard for any other reason, it hasn’t been “destroyed.”
701.6c A regeneration effect replaces a destruction event. See rule 701.12, “Regenerate.”
Bury (Obsolete) - A term that meant “put [a permanent] into its owner’s graveyard.” In general, cards that were printed with the term “bury” have received errata in the Oracle card reference to read, “Destroy [a permanent]. It can’t be regenerated,” or “Sacrifice [a permanent].”
(Note: The only way to cancel a mana ability is due to a player making an illegal action (i.e. Casting a spell, but realizing that they don't have the mana (or other additional costs) necessary to pay for the spell. In this situation, the game is backed up to a previous state just before the illegal action. Mana abilities used to pay for the spell are reversed.)
Handling Illegal Actions
717.1. If a player realizes that he or she can’t legally take an action after starting to do so, the entire action is reversed and any payments already made are canceled. No abilities trigger and no effects apply as a result of an undone action. If the action was casting a spell, the spell returns to the zone it came from. The player may also reverse any legal mana abilities activated while making the illegal play, unless mana from them or from any triggered mana abilities they triggered was spent on another mana ability that wasn’t reversed. Players may not reverse actions that moved cards to a library, moved cards from a library to any zone other than the stack, or caused a library to be shuffled.