Can you tap a creature for mana using a mana ability the turn it enters the battlefield even with summoning sickness?

Summoning sickness (CR 302.6) prevents a creature from attacking or being tapped to pay the cost of an ability that uses the tap symbol ({T}) unless you have controlled it since the beginning of your most recent turn. This restriction applies regardless of whether the ability is a mana ability or any other kind of ability.

A mana ability is defined in CR 605.1 as an ability that could produce mana and doesn't require a target. Being a mana ability does not grant any special exemption from the summoning sickness rule. The exemption that exists for mana abilities (CR 605.3b) is about timing — mana abilities can be activated any time mana is needed — not about summoning sickness.

The only way around this is if the creature has the Haste keyword (CR 702.10a), which explicitly allows the creature to attack and be tapped as though you had controlled it since the start of your turn, even on the turn it entered the battlefield.

Example: You play a Llanowar Elves on your turn. You cannot tap it for {G} that same turn because you haven't controlled it since the beginning of your turn. If it instead had Haste (like a Zhur-Taa Druid), you could tap it for mana immediately.

In summary: summoning sickness blocks the {T} cost universally. Haste is the only standard override. Plan accordingly by playing mana creatures before the turn you need to use them.