As Nikolay pointed out, one can amuse oneself looking for new meanings of the old words, or new words, or some such, but this is hardly a specific characteristic of the communist-era influence. Any language develops, acquires new words/meanings etc. Yes it can be argued that the Bolshevik revolution forced some of these changes (Anixx mentioned some), but I don't think it's important. I guess, the French revolution brought more changes (in this sense) to the French language than the Russian revolution to the Russian.
It would be more interesting to consider formal or conceptual changes in and around the language. They are invariably influenced by the culture, on which the revolution had a profound impact. I can think of a few.
- Love of acronyms and abbreviations
Нардеп (народный депутат),
Реввоенсовет (революционный военный совет),
Минсредмаш (министерство среднего машиностроения),
ВАЗ/ГАЗ/КамАЗ (Волжский/Горьковский/Камский автомобильный завод), or just such word contraptions as
Востокнефтегазэлектромонтаж. They were everywhere and were often so integrated that not only they followed normal grammatical rules, but even their derivatives sounded normal (минсредмашевское заседание, ВАЗовец).
In the 1920s, it was even common to invent names in a similar manner.
Даздраперма (да здравствует Первое мая, a female name) is a notorious example.
George Orwell spotted this trend perfectly. He notes somewhere in 1984: words like
Ingsoc acquire their own attached meaning, which may end up being substantially different to what it would be if you try to consciously break it into parts. But this is the trick: you never do. Never need to. It doesn't even cross your mind (unless you make an effort) that this is English Socialism (wait, why English? and what exactly is socialism?) Orwell didn't invent the phenomenon (just like most social aspects in 1984); he studied the USSR.
- Governing body for the language
The Soviets established a body (now called Институт русского языка) that had a mandate to establish definitive rules of the Russian language. It issued the complete 'codex' of the orthographic rules (at the time of collapse of the USSR, the latest one was from 1956!), grammar, etc.
This is not uncommon; France and Germany have similar bodies. But I'm not aware of a similar establishment in tsarist Russia.
Centralisation is a common Russian trait. For the language, it partly relates to the previous point. Not only the formal rules of writing, but even pronunciation was standardised (mostly around the Moscow dialect). Local deviations were often considered 'improper' language.
Certainly, radio and TV played their role, and they can hardly be classified as Soviet inventions. Nevertheless, one could rarely be found to be 'proud' of his dialect (the way many Europeans are); it was something to be ashamed of. Someone lucky enough to move to Moscow would strive to learn to speak in a Moscow way, so as not to look provincial.
Of course, dialects existed, and professionals studied them. But at school, kids were taught (or supposed to be taught) the 'proper' language across the country. (That said, Russian is remarkably uniform compared to most other European languages).
This is somewhat hard to prove as there are no recordings, but I read that in the 19th century it was the norm amongst the educated people to have a rolling R, similar to the French one. This, apparently, had to do with the French, which Russian aristocrats often learned as the first language. More generally, the French influence was enormous, perhaps greater than the English has now.
After the revolution, this was all gone rather quickly.