Generally speaking, there is a certain distaste, to put it no more strongly, for exhumations unless they are absolutely necessary, and even then.
I don't know about the position in Russia, but in England & Wales (Scotland has a separate legal system which I don't know much about) a court order is required and they are not easy to get. Exhumation orders are granted from time to time where there is a suspected murder, the victim has been buried and exhumation is needed to get evidence for the investigation. Another example is a scientific investigation into Spanish 'flu, which was carried out about five years ago (the BBC did a programme about it. The politician and diplomat Sir Mark Sykes died from Spanish 'flu during the Paris Peace Conference and was buried in a lead coffin. The investigators hoped that his remains would be sufficiently preserved to enable them to extract the bacterium. They got an exhumation order, but a major factor was that Sir Mark's family were very happy for it to go ahead (the present baronet said in the programme that his grandfather had been a very public-spirited man and if he could have approved, he would have!). Unfortunately, the coffin had spilt and they didn't get anything useful, but the point is clear.
I doubt whether the current Romanovs would present a sufficiently united front to override the general presumption against exhumation. Getting an exhumation of Georgy Alexandrovich was apparently pretty difficult.
Ann