If the creature doesn't have haste then the answer is no (if you have just summoned it that it) Otherwise if you have controlled it for at least one full turn then yes you may indeed activate an creature's tap ability during the opposing turn.
Creatures may not attack or use abilities that require them to tap until the beginning of their controller's next turn. This is commonly called "summoning sickness".
@DukeNicky: Actually, a creature only loses summoning sickness if you have controlled it continuously since the beginning of your most recent turn (not necessarily one full turn).
If you cast a creature on your turn and it does not have or gain haste, then you can't attack with it or activate any of its activate abilities with either a tap or untap symbol in its activation cost until your next turn begins.
There's a distinction between an ability that has the tap symbol and an ability that requires tapping. If I'm not wrong, you can tap Azami, Lady of Scrolls for her own ability the turn she comes into the game
That is correct. Summoning sickness only interferes with abilities that have the tap or untap symbol in their activation costs.
DaShPrime is absolutely correct. To clarify, if you look at Raging Goblin for example. His haste says can attack and (tap symbol) on the first turn he's played. That is ALL summoning sickness restricts. Lullmage Mentor 's second ability, for example, says the word tap, so you can do it with merfolks that just entered, such as ones from his ability.
@DaShPrime: The tapping of the creatures represents a COST and not a "tap ability" itself. Much like Grand Architect 's ability can be used the turn it comes into play as the tapping of creatures is a COST.
Also, you cannot tap a creature or attack with it UNLESS you begin your turn with said creature on your side of the battlefield (obviously ignoring haste in this instance)
@harrydemon117: Your explanation does not make sense. There is no rules distinction or definition that backs up what you're saying. All activated abilities have activation costs; abilities with tap symbols in their activation costs and abilities with the word tap in their activation costs are both examples of activated abilities that require something to be tapped as a payment.
Epoch, your earlier answer said you cannot use abilities with the untap symbol - but haste does not mention that.
Is using untap abilities allowed with summoning sickness (Assuming your creature is tapped somehow), or is it just that haste does not remove that restriction?
Directly from the Comprehensive Rulebook:
302.6. A creature's activated ability with the tap symbol or the untap symbol in its activation cost can't be activated unless the creature has been under its controller's control continuously since his or her most recent turn began. A creature can't attack unless it has been under its controller's control continuously since his or her most recent turn began. This rule is informally called the "summoning sickness" rule.
Why doesn't it make sense? The tap ability is a cost of the creature being able to tap itself whereas the statement "Tap target creature you control:" is a cost YOU pay and is not dependent on the creature being able to tap itself
@harrydemon117: I understand what you're trying to argue, but both the premise and the wording are wrong. You grasp the concept of summoning sickness, clearly, but you may not understand the reasoning properly.
Because the phrase "tap ability" is too vague. "Tap ability" can refer to an ability with the tap symbol (Prodigal Pyromancer ) or to an ability that requires you to tap something (Azami, Lady of Scrolls ). Only the former is shut down by summoning sickness, but you don't clarify the distinction. Furthermore, the phrase "tap ability" is not used in the CR for this exact reason. Instead, the CR uses the wording "a creature's activated ability with the tap symbol or the untap symbol in its activation cost" to make the distinction clear.
Also, your focus on the cost is irrelevant because both kinds of tap abilities (Prodigal Pyromancer and Azami, Lady of Scrolls ) require tapping as a cost. The cost isn't the important part because it isn't a distinction between the two things you're comparing. The tap symbol is the difference.
In regards to your reply (post #11), your wording is still unclear. You say "the tap ability is a cost of the creature being able to tap itself," which doesn't make sense. Abilities are not costs, and this statement doesn't clarify your argument. Additionally, the foundation for your argument is flawed. There is no distinction between a creature "tap[ping] itself" and the player tapping it because creatures don't tap themselves. The player is the one who taps the creature either to pay a cost or as part of an effect. In this case, both instances (Prodigal Pyromancer and Azami, Lady of Scrolls ) are costs that the player pays to activate an ability.
In fact, the reason that both the tap symbol and the phrase "tap an untapped X you control" exist is because the tap symbol is used in cases that only involve the source permanent. Prodigal Pyromancer 's ability can only be activated by tapping Prodigal Pyromancer . Azami, Lady of Scrolls can be activated by tapping any Wizard you control, not just Azami herself.
This is just nitpicking: costs don't target. You will never see the cost "tap target creature you control." The effect is what targets, and effects are irrelevant for the purposes of summoning sickness.
The second paragraph should begin, "In response to your first post:"
I don't know why that got cut off.
I understand. I understood the reasons for each but I wasn't able to articulate this :)
This is why I will be a player and not a judge haha