Is playing online with non-believers—such as playing chess—considered showing loyalty to them, based on the fatwa that prohibits intermingling with them in food and playing with them?

Merely playing with disbelievers online is not considered the prohibited form of al-muwalat (showing loyalty/allegiance), unless it is accompanied by loving them, relying on them, or feeling at ease with them. One must be cautious of loving them, relying on them, taking them as close confidantes, showing cheerfulness and friendliness towards them, feeling at ease with them, honoring them, and drawing them near.