Can a creature with summoning sickness activate a tap ability that doesn't have the tap symbol in its cost?

Summoning sickness (formally called the 'attack and tap' restriction) prevents a creature from attacking and from activating abilities that include the tap symbol ({T}) or untap symbol ({Q}) in their activation cost. This is defined in CR 302.6.

If a tap ability's cost does not use {T} or {Q}, the summoning sickness rule simply does not apply to it. The creature can activate that ability freely on the same turn it entered the battlefield, even if it's under your control and hasn't been under your control since the beginning of your most recent turn (CR 302.6).

For example, imagine a creature with the activated ability '{2}: Deal 1 damage to any target.' Because that cost contains no {T} or {Q}, summoning sickness is irrelevant—you can activate it the moment the creature enters the battlefield.

Contrast this with an ability costing '{T}: Draw a card.' That ability cannot be activated by a creature with summoning sickness, because the {T} symbol is explicitly in the cost (CR 302.6, 106.1a).

In short: check the activation cost. If {T} or {Q} appears there, summoning sickness blocks it. If neither appears, the creature can activate the ability regardless of how recently it entered the battlefield.