I have a DX7 MkI. I am concerned about the noise. I know that the first DX7 version uses 12 bit converters that can produce lot of noise, but it gets noticeably worse when using high shelving filter. I am aware that high shelving filter boosts everything above including the noise, still it seems too much...

I recorded a small sample of famous E. piano 1 preset, no processing applied, recorded directly into the preamp and high quailty AD converter.

There is no high shelving Filter in the original DX7 (FM synthesis did not include any Filters in the DX7)... So you are applying processing if this filter is in place.

Original DX7 FM sounds can be loaded directly to the current MONTAGE/MODX.

If you mean the using the external high shelving filter, then there's no reason to set it so that it emphasizes high frequencies. It must be set opposite curve, to suppress high frequencies. Only that way it will suppress also the noise.