Yes, certain restrictions are well known. One reference is Kuperberg & Schramm here which says that such graphs would have to have average degree <15. A more recent reference is Benjamini & Schramm here which shows that certain low degree infinite graphs are not realizable this way.

Yes, certain restrictions are well known. One reference is Kuperberg & Schramm here ("Average kissing numbers for non-congruent sphere packings", 1994) which says that such graphs would have to have average degree <15. A more recent reference is Benjamini & Schramm here ("Lack of Sphere Packing of Graphs via Non-Linear Potential Theory", 2009) which shows that certain low degree infinite graphs are not realizable this way.