I'm going to directly address I Love Lucy and skip most of the "what if" part if that's alright - This show is historically credited with "inventing the rerun" during Ball's pregnancy and that clearly demonstrates there was no thought put towards reruns at the show's inception.
"Prior to I Love Lucy rerunning its episodes during the summer, shows typically went on a summer hiatus and were replaced with summer replacements, generally lower-priority programs"
Remember, the first television sitcom was broadcast was in 1947 and it's listed in the Guinness book of world records as Mary Kay and Johnny, the first ILL was 1951, that's only 4 years apart. During the broadcast run, there was no question that the show was made between the years 1947 and 1957 if you were watching ILL during its original broadcast (1951-1957).
Moreover, the first "talking picture" was 1927, the first television broadcast was 1928, so this contextually narrows the window for when it was made even if you (as a person of that era) were not well versed in the "new technology".
I think the question, the primary part of the question, is valid. Did they ever mention the year? No, probably not, they did not have to and there really wasn't any reason to because television "was in it's infancy", it dated itself. Also, mention of politics on an entertainment program was probably highly frowned upon - remember that this is the era where also during 1947 the first Mccarthy era Hollywood blacklist appeared and it took down a lot of prominent people, Even Charlie Chaplin was labeled a communist in 1952, one year after ILL began broadcast.
To very briefly address the controversy which for some reason has manifested around the Roman numeral theory, I personally can tell you that if you flash a ton of Roman numerals quickly in front of me I might not be able to read them because once you get out of the tens or hundreds position it all starts to bleed together. The first commercially available VCR the Sony U-matic didn't become available in Japan until 1971 and without an ability to pause live tv (with a VCR, DVR, ETC), the average person might not "on the fly" know that MCMLVII was read as 1957, given that in 1950 3.2% of people in the United States over 14 could not read or write in ANY language. Certainly outside of a liturgical training, private education or institutes of higher education people were very unlikely to be well versed in Latin. It was not more likely that someone in the era of interest would be able to read Latin or Roman numerals than it is today.
There was a practice in film and television of not showing "modern" cell phones (which is why for a block of years you see actors using the same outdated flip phones) which is similar to the suggested use of Roman numerals as production year obscura - to not immediately hint that something a person is watching is a rerun - but as you can tell from current film and television where there are entire plots surrounding smartphones - they have abandoned the practice.
Given the facts, it's safe to conclude I Love Lucy was not intentionally obscuring the year of production so as to secure itself as a re-runnable property, but that doesn't mean they went out of their way to mention it either.
I think this is a valid question, I hope you feel this is a valid answer.
Incidentally, by searching google for only a few seconds I found this
link which directly references episodes of "the lucy show" which also starred Lucille Ball in the 1960's and several episodes feature plots surrounding the world's fair. The world's fair in the United States is very easy to date because it is not held in the US every year. The evidence is there, it is up to you to put together the pieces.
You may, however, find that these are not the droids you're looking for.