I heartily agree with all of the theoretical points made by @user6726 (+1). This is a great question (+1) because it illustrates a common misconception about what lexical tone is and how it functions in language.
WHAT LEXICAL TONE IS NOT
Tone is not intonation; that is, the function of tone in a language like Mandarin is not to convey information about whether the sentence is a question or a statement, or to provide "paralinguistic" information about the state of mind of the speaker. Tone is not a means of providing "color" to the speaker's speech. Tone is also not predictable. In a tone language, there's no way to determine what tone a syllable or word should have based on other phonological properties of the syllable/word, such as its segmental (vowel and consonant) makeup.
WHAT LEXICAL TONE IS
Tone is a phonological feature on words and syllables that is as integral to the meanings and grammatical functions of those words and syllables as the segments (vowels and consonants) that make up the words and syllables. The lexical tonal pattern for a word or a morpheme must be learned and memorized as part of the mental dictionary entry for that word/morpheme. And since it is part of the dictionary entry, tone is language-specific.
Once you internalize the concepts above, you should start to see why it doesn't make sense to think of imposing the lexical tonal system of one language onto a different language that doesn't have a lexical tone system.
Let's delve a little deeper into your thought experiment with Russian and Mandarin. Tone in Mandarin is assigned at the level of the syllable, and syllables in Mandarin roughly correspond to morphemes (units of meaning). For a given syllable in Russian, how would you decide what tone that syllable should be assigned? Would you try to do it by meaning? You wouldn't get very far trying to find the equivalent Mandarin morpheme for every Russian morpheme, because there would be many untranslatable morphemes, many morphemes with multiple possible translations, and many polysyllabic morphemes. On the other hand, you couldn't do it by sound, either, since Mandarin is filled with sets of syllables that have the same segmental makeup but different tones. Not to mention that Russian has many syllables that have no straightforward phonetic equivalents in Mandarin.
It might be tempting to draw parallels between what you're describing and speaking one language with the "accent" from another language--as in speaking English with a French accent. But there's a crucial distinction that renders such an analogy specious: speaking English with a French accent involves making substitutions of one segment for another--mapping the French vowel space onto the English vowel space, substituting [z] for [ð], replacing the intonational contours of English with those of French. But Russian has no lexical tonal system, so there's nothing there to substitute with Mandarin tones.
OK. So what could you do?
- You could invent a tonal system for Russian from scratch. This is what @guifa and @user6726 seem to be suggesting. You'd have to decide to what unit the lexical tonal patterns would apply in Russian, and you'd have to assign a pattern to every single word in the Russian dictionary. I agree with @user6726 that the normal speaking rhythm of Russian doesn't lend itself to a Mandarin-style system, so I'd go for something more along the lines of the systems found in languages like Yoruba or Japanese, where many words are polysyllabic and thus the tonal pattern, made up of a series of relative high and low pitches, is assigned to the word. One of the most complicated aspects of the tonal system that would need addressing is how the lexical tones in this new version of Russian would interact with the already existing intonational system of the language, since both tone and intonation would dictate the pitch patterns present in the complete utterance. Tone-intonation interactions are completely language-specific and unpredictable, so you'd have to decide, for example, how the tone on a given word is produced when the word is part of a yes-no question versus when it is part of a declarative statement.
- You could manipulate Russian utterances such that the syllables in them were characterized by durations and pitch contours typical in Mandarin. I can't stress enough, however, how much this wouldn't be "speaking Russian with Mandarin tone". This would basically just be like singing in Russian--imposing melodies and rhythms on the words and phrases--and the melodic contours would not be contributing anything lexical or grammatical to those utterances. There would be several ways to achieve this effect. For my money the most straightforward way would be to find a native Russian speaker who is fluent enough in Mandarin to take a crack at it. I know they're out there, because when I took Mandarin in grad school there was a Russian student who was learning Mandarin in order to be able to speak the native language of his future wife, who was Chinese! If I were to try to use speech technology to generate such utterances myself, I wouldn't use outright speech synthesis; rather I'd take recordings of carefully articulated Russian and use Praat's manipulation functions to alter the durations and fundamental frequency contours of all of the syllables. For both the Russian human and for me, the decisions about what contour to give to each syllable would be completely arbitrary (other than, perhaps, avoiding too many strings of the same contour in a row and also avoiding sequences that don't generally occur in Mandarin).
- You could forget the Mandarin-Russian example and focus instead on two languages that both have lexical tone. In this way, you could bring the task more in line with the English-with-a-French-accent analogy. Ideally you'd want two languages that have similar tone-bearing units, like the syllable in both or the word in both. You could then come up with a straightforward substitution rubric--for Tone 1 in Language A replaces Tone 3 in Language B, Tone 2 in Language A replaces both Tone 1 and Tone 5 in Language B, etc. In such a scenario, you'd also have to decide what to do about that pesky tone-intonation interaction. The easiest thing to do would be to adopt the intonation of Language A along with its tones.