Can you tap a creature for mana the same turn it enters the battlefield if it has a mana ability?

By default, a creature cannot be tapped to activate an ability — including mana abilities — if it has been under your control since the beginning of your most recent turn. This rule is commonly called "summoning sickness" and is found in CR 302.6: a creature cannot attack and its activated abilities with the tap symbol (or untap symbol) cannot be activated unless it has been under your control continuously since the start of your most recent turn.

Mana abilities are not exempt from summoning sickness. CR 302.6 applies to any activated ability that uses the tap or untap symbol in its cost, regardless of whether it produces mana. So a Llanowar Elf that enters the battlefield on your turn cannot tap for mana that same turn.

The exception is Haste (CR 702.10). A creature with Haste is explicitly exempt from the summoning sickness restriction of CR 302.6, meaning it can attack and can tap its abilities — including mana abilities — the same turn it enters the battlefield.

Concrete example: You play a Llanowar Elf on your main phase. You cannot immediately tap it to add {G} to your mana pool because it just entered the battlefield and does not have Haste. If instead you played a Goblin Anarchomancer (which has Haste), you could tap it for mana immediately after it enters.

Note: This restriction applies only to activated abilities using the tap/untap symbol. Triggered or static abilities on a creature (like those that trigger "when this creature enters the battlefield") are unaffected by summoning sickness and work normally the turn the creature enters.