Let's start with: you cannot Tribute a monster for no reason - say, if all your Monster Card Zones are occupied.
You cannot Tribute a monster for a Tribute Summon the turn it is Summoned, because this would violate the "1 Normal Summon or Set per turn" rule. If you have Ultimate Offering active and at least 501 Life Points, this would be allowed because you can pay 500 Life Points for the extra summon.
You can Tribute a monster for the cost of a card effect during the same turn it is Summoned. For example, you can Tribute your monster for Catapult Turtle's effect if your monster was just summoned.
If the monster has an effect where it would Tribute itself (like Exiled Force) - like with any elective monster effect, you have an opportunity to do this before your opponent can activate cards like Trap Hole. You must surrender priority to your opponent if you do not wish to do this.
Generally, people do not announce priority switches. This is a problem because your opponent's Trap Hole would cause you to miss your opportunity to Tribute it (you would not be able to Chain because your effect is Spell Speed 1 and Trap Hole is Spell Speed 2). And you would not be able to say "Hold on, don't be so quick, I wanted to use its effect!", because maybe you are lying and you only want to do that now that you saw the Trap Hole.
In short, it depends what the Tribute will be used for.
You cannot Tribute a monster for a Tribute Summon the turn it is Summoned, because this would violate the "1 Normal Summon or Set per turn" rule. If you have Ultimate Offering active and at least 501 Life Points, this would be allowed because you can pay 500 Life Points for the extra summon.
You can Tribute a monster for the cost of a card effect during the same turn it is Summoned. For example, you can Tribute your monster for Catapult Turtle's effect if your monster was just summoned.
If the monster has an effect where it would Tribute itself (like Exiled Force) - like with any elective monster effect, you have an opportunity to do this before your opponent can activate cards like Trap Hole. You must surrender priority to your opponent if you do not wish to do this.
Generally, people do not announce priority switches. This is a problem because your opponent's Trap Hole would cause you to miss your opportunity to Tribute it (you would not be able to Chain because your effect is Spell Speed 1 and Trap Hole is Spell Speed 2). And you would not be able to say "Hold on, don't be so quick, I wanted to use its effect!", because maybe you are lying and you only want to do that now that you saw the Trap Hole.
In short, it depends what the Tribute will be used for.