I've done a bit of reading and I can't find any details as to a possible cause of death; I read that she suffered from dizziness and refused to be examined by a doctor, but that's it. Anyone know more? Was it possibly connected to her having never had children (i.e. some kind of reproductive system issue or cancer)? Would be really curious to learn more.
According to Simon Sebag Montefiore's book The Romanovs, 1613-1918, Elizaveta suffered a stroke on 23 December 1761 (Old Style). It must have been a seriously debilitating stroke because she fell unconscious and died two days later, at 4 PM on Christmas Day. For this information, Montefiore cites the memoirs of Catherine the Great, niece to Elizaveta through her marriage to Peter III, Elizaveta's nephew.