Which xiangqi pieces can never give check to the enemy general, no matter what?
These answers restate the rules implemented by our open-source engine, which follows the WXF laws for movement, checkmate and both repetition rulings: a side that forces a repetition by perpetual check or perpetual chase loses rather than drawing. The chase half has not been validated against the official casebook. Where an answer refers to a specific position, that position lives on the linked reference page, where the build verifies it before publishing.
Direct answer
The advisor, the elephant, and the general itself. The advisor and elephant are confined by their own movement rules to their own side of the board and can never come anywhere near the enemy palace. The general never checks either: two generals facing each other on an open file is a position that may never arise, not a check. Only the chariot, horse, cannon and soldier can give check.
Why this is worth knowing while calculating
When you are working out what could possibly check your general next move, you can rule out the opponent's advisors, elephants and general immediately and completely, not just "usually" or "probably." Nothing about the position, however unusual, changes that; it is a fixed property of how those pieces move, not a situational judgement.
What they contribute instead
Their entire value in an attack is indirect: removing your own defender near the palace, or supporting another piece's threat, never delivering the check themselves. Every mating pattern in xiangqi that appears to involve an advisor or elephant is really the advisor or elephant enabling a different piece to deliver the actual check.
Related questions
Is the flying-general rule an exception that lets the general check?
No. It forbids the two generals facing each other on an open file; any move that would create that position is illegal, so it is never a check.
Can an advisor or elephant ever check its own general by mistake?
No, self-check from those two pieces is impossible for the same reason, they can never occupy a square that threatens either general.