Do you exhaust Forgotten Signpost after use?

Yes, you do exhaust the Forgotten Signpost to activate its ability.

According to its card text and the Riftbound rules:

• The card has the [T] symbol (Exhaust symbol) in its ability cost.
• Rule 414.5 states: "In abilities, the Exhaust symbol represents the cost 'Exhaust this' or 'Exhaust me.'"

Because exhausting the card is a required cost to activate the ability, the Forgotten Signpost must be exhausted as part of paying for that action.