It remains open whether the won-position problem of infinite chess is decidable, the problem of determining whether a given finite position in infinite chess is winning for white or not. See Richard Stanley's question at https://mathoverflow.net/q/27967. Meanwhile, the mate-in-$n$ problem of infinite chess is decidable, and so the open part of the problem necessarily concerns positions with infinite game value.

Also open is the optimal-play problem: given a won position, is a given move optimal?