You might get better answers to this question over at /r/linguistics
As a rule, Nope.
Certain things do simplify over time, but others go the other direction.
One of the most common causes of this is people from outside the initial immediate language community coming in and adopting the language, thus "messing it up" as some people might describe it. As an example from Sinitic: Shanghainese has a significantly smaller tone inventory than the neighbouring dialects of the same language (Wu). Most Wu dialects have 7 tones. Shanghainese has 5. The main cause of this is the fact that Shanghainese speakers have their origins in various other dialect speaking areas. Original Shanghainese speakers (original to lets say 200 years ago) mixed with Suzhou dialect speakers and Ningbo dialect speakers, all who had slightly different tones, and that mixture led to a reduced inventory.
Additionally, languages in contact tend to converge. In the Balkans, you have what's called the Balkan sprachbund, an area where unrelated languages now share common features. You can think of these features as being contagious. So Albanian and Romance languages and Slavic languages, while all different branches of Indo-European, have in many ways converged and become more similar to each other over time. From one perspective you can call this simplifying, though if you were a speaker of a closely related language to which this convergence did not happen, say it was isolated from this for whatever reason, then you may not consider it simplification.
But languages also get more complex over time. Again due to convergence, look at Assamese spoken in India. It's an Indo-European language, but has a fully formed Tibeto-Burman classifier system. This is because for centuries it was in close contact with Tibeto-Burman and Tai-Kadai (Thai) languages, both which have these. In that regard, from the perspective of all the other Indo-Iranian languages, Assamese has become much more complex. However it has also lost the retroflexes found in other Indo-Aryan languages. (I think Bengali has as well, to which Assamese is closely related, but don't quote me on that.)
Think of it this way:
All languages have to be equally good at encoding human thought for communication. Those that aren't will be changed by the speakers (see native second-gen Esperanto speakers as an example). This means there's a minimum required level of complexity.
All languages are also limited in how complex they can be because otherwise communication efficiency suffers. In other words there's also a maximum level of complexity a language can have.
All languages have a certain level of redundancy built in. Think of this in terms of signal to noise ratio. Redundancy allows communication to work even in incomplete transmissions of data, i.e. talking in a crowded noisy room or a heavy rainstorm in the jungle.
Languages which violate this in either direction will be changed by speakers to make up for it, either by simplifying needlessly complex things, or by adding complexity to clear up ambiguity.
When linguists say (as we often do) all languages are more or less equally complex as all other languages, this is what we mean: You can't go too far in either direction, and given another generation, all languages with speaker communities will auto-correct.
As an example of simplification necessitating complexity, look at Chinese. Old Chinese didn't have tones, but had a much more complex phonemic inventory. With a reduction in the number of consonants available would come an increase in ambiguity, however this then came with the introduction of tones, which then preserved the distinctions and prevented ambiguity. In fact tones probably came slightly before the reduction in consonants, but they were essentially co-incidental; One wasn't happening without the other.*
If you take the idea of simplification a few steps further, we'd all be grunting and speaking with incredible simplicity, or in the other direction, all our ancestors must have spoken the most complex language imaginable. Neither are the case, because there are limits in both directions.
The only thing that operates as a rule is that change is always happening. Not always in just one direction or in just one area of language, and not always at the same rate, but it's always there.
* Just in case anyone wants to take issue with this and pick nits, I should add that this only was the case because the structure the syllable could take was already fairly restricted, and tonogenesis wouldn't happen as easily with something like English where you have much much more complex syllables. Tonogenesis was easy in places like East/Southeast Asia because the other pieces were already in place that helped it along. That's why even those varieties of English which do have phonemic tone have it only with a very low functional load where it's still almost not important.