A paper in Diplomatic History titled "From Disarmament to Rearmament: The United States and the Revision of the Italian Peace Treaty of 1947" covers the issue fairly well. The bottom line is that as the situation changed immediately post-war, and Italy was proposed for membership in NATO, it became quite clear to the Western allies that there was a conflict between the treaty and Italian membership in NATO. This led to a variety of discussions about just what to do about it which is covered by the article.
One action that the Soviets took was to block Italy joining the UN until 1955.
However, with respect to the arms limitation, the paper notes:
If the Soviets were to protest when Italy went beyond the limits, the Western powers would reject the protest and refer to Soviet satellite treaty violations
That is, the Soviets had already armed other treaty countries under its control over the Treaty limits.
On 26 September 1951 the United States, Great Britain, and France, in a tripartite declaration, publicly expressed their willingness "to give favorable consideration" to an Italian request for the removal of certain restrictions...
The Soviets replied to the tripartite declaration on 11 October. In return for easing the treaty restrictions and allowing Italy in the United Nations, the Soviet Union demanded that Italy leave NATO and bar foreign military bases...
The United States responded by reaffirming its position and noting that one reason for seeking to revise the Treaty of Peace "was the Soviet Satellites had not remained within the military limitations imposed on them."
On 6 February 1952 the Soviet Union vetoed an Italian application for UN membership for the fifth time. On 9 February the Italian government informed the Soviet ambassador to Rome that Moscow had violated the peace treaty by its veto. Therefore, Italy no longer considered itself bound by obligations to the Soviet Union imposed by the treaty.
In reality there never was a revision of the treaty. The Western powers simply authorized the Italians to rearm.
So, yes, the Soviets threw up roadblocks along the way, but ultimately the world moved on to the new reality of NATO vs the Warsaw Pact. In 1952, any restrictions on Italian re-arming were moot.
Per Wikipedia it wasn't until December of 1954 that the first submarine (USS Barb, SS-220) was transferred to Italy and renamed Enrico Tazzoli (S 511), where it remained in service until 1972. So, while the Soviets tried to stop the possibility of Italy re-arming, I'm not sure they actually complained about the delivery of submarines to Italy, occurring several years after the Treaty was, in essence, overcome by events.