However, if by evolve we mean "to change in a Darwinian sense"--from initial simplicity to later complexity--that the answer is that there is no clear, unequivocal evidence of such an evolution of languages. To illustrate, I will use two languages that I am fairly familiar with--Modern Chinese and Classical Chinese. The former I have studied since my college days, while the latter I begin to study in more recent years. Classical Chinese refers to the written form of Chinese as it existed starting in the fifth century BC and continuing until the third century AD (subsequent to the third century, a modified form of classical Chinese, known as Literary Chinese, continued to be used for written Chinese until the early 20th century--although it differed from the spoken language). However, while there is a gap of over two millennia between Modern Chinese and Classical Chinese, there is no evidence of a Darwinian-style evolution from simplicity to complexity between the two forms of the language. Yes, there is a clear evolution in the broadest sense--for example, certain aspects of the grammar have changed over the years. To illustrate, Classical Chinese often omitted subjects in sentences, but this is less common in Modern Chinese. Nevertheless, Classical Chinese is in no way less complex than Modern Chinese. It is true that Classical Chinese is more concise than modern Chinese--that is, Classical Chinese tends to express ideas in fewer words than Modern Chinese. For example, whereas in Modern Chinese a conditional sentence is introduced with a word equivalent in meaning to the English word if, in Classical Chinese it was possible to place one clause in a conditional relationship with another clause without using such a word--it was the context that implied a conditional relationship. Now, we might think that the lack of a word signaling a conditional relationship is evidence that Classical Chinese was less complex than Modern Chinese, but this is not necessarily so. It simply means that Modern Chinese is more explicit about logical relationships between clauses. Which is more complex--a language that implies certain logical relationships, arguably requiring more mental effort on the part of the reader--or a language in which such relationships are spelled out? It is difficult to say.
In short, the answer to the question posed at the beginning of this post is: "Yes, and no." If by evolve we simply mean "change," then yes, there is clear evidence that languages do evolve. If, on the other hand, by evolve we mean "change from relative simplicity to relative complexity," then the answer is "no"--there is no actual evidence of languages changing in a Darwinian sense. This fact has not kept many linguists from stating that the human language in its earliest form must have been much less complex, but such an assertion is mere speculation, arguably based more on a commitment to Darwinism than the actual evidence we have.
Image of ancient Chinese calligraphy from Wikimedia Commons