Apparently (unconfirmed) the very first Beta/VHS release back in 1982 is a rare uncut version that is different from all other home video releases which were slightly cut. Here is what I know:

The uncut version of Dr. No shows a more extended fight with the taxi driver, Bond shooting more bullets at Professor Dent and Dr. No’s henchmen beating up Bond was toned down.

all home video releases afterwards (VHS, Laserdisc, DVD, Blu-ray, etc.) include the following cuts:

Dr. No’s line I’m sure she will amuse the guards was considered too sexually suggestive, and has been redubbed instead to The guards will amuse her.
Shortly after, Dr. No’s henchmen beating up Bond as he is sat at the table has been trimmed, introducing a dissolve to the next scene much earlier to cover the removal of violent footage.
Bond’s fight with the chauffeur has been trimmed, including the removals of blows to the chauffeur when he is unable to defend himself. A knee kick in this fight was also removed.

In 1983 there was a re-release of Dr. No via CBS/Fox on CED, Beta, VHS and Laserdisc. This leads me to believe they “fixed” something I mean why else would they re-release the movie just one year later?

I have checked this against the very first laserdisc release from 1982 but it has all of the cuts mentioned above.

The release in question would have one of these two covers:

(source: http://007homevideo.com)

Unfortunately I have been able to get a hold of one these tapes. Hopefully someone may have better luck and we’ll be able to confirm this.