No. Plasmids should generally be considered as a circular DNA rather than linear. However, no matter where you cut it, the positions don't change.
The position of the cut site is still the position on the plasmid at which the RE cuts. The numbering on plasmids is based on the position of a common restriction site, which depends a bit on which plasmid you are using. For pBR322 based plasmids, it is in the middle of an EcoRI site. This is because when plasmids were first discovered and used, we didn't have sequencing so the sequences were determined by restriction fragment mapping. Thus the start site was determined based on the position of a cut site in one of the restriction enzymes.
The positions of your primers are relative to the sequence you are inserting - if you are using something like SP6/T7 from the plasmid, those are sites on the plasmid and would be noted as such at the relevant sites when publishing the sequence. If you are sequencing off sites internal to your insert, then you would add on the bases of the plasmid when reporting these sites. You can also report the bases position on the insert; as an example:
We inserted gene ABC into plasmid pBR322 between HindIII and EcoRV and sequenced using primer specific to insert ABC, at position XX on the plasmid (YY on the insert).
However, if you take your linear (cut) sequence and put it into a sequence manipulation program (e.g. Snapgene, Lasergene, Genious, etc. (no affiliations or recommendations for any of those, just commonly used ones)), then it might annotate your plasmid as having position 1 at the cut site, but this is an artificial annotation based on what you have told the program. If you circularize the sequence, the program may well recognize the plasmid and annotate the sites of interest (e.g. resistance markers, MCS, Ori) correctly with their proper positions marked.
Edited to add: Some positions on the plasmid do change once you have put an insert into it. These are the positions after the insert. For example, let's say you have a (hypothetical) plasmid of 500 bp in length, which has a promoter at position 50 and a resistance marker starting at position 300. If you insert a 100 bp fragment at position 250. The plasmid is now 600 bp long, the position of the promoter is still 50, but the resistance marker now starts at 350.