Does language learning improve or worsen the ability to remember and process information? In an interview with a Czech neurologist Syka, I've heard that much of the mental health (such as the ability to reason, communicate and process information) in older age mainly depends​ on the amount of informational ballast we create. The argument states that with this additional information, we unnecessarily wear some of the basic mental mechanisms out (Even though there's no the effect of "too much memories"). Does knowledge of more languages tend to improve or worsen these mental skills?
What I do not quite understand is: What is (according to Syka) "useless information" and how should we "avoid" this kind of information? 

Let me try and answer this as follows: During our lifetime, our brain undergoes "synaptic pruning", which lasts from childhood into puberty. Basically, the brain gets rid of "unused" synapses to make space for more complex structures according to the principle of "use it or lose it". It has been hypothesised that sleep has a similar function in that it helps decrease the strength of synapses in order to save energy and not store what you may have referred to as "information ballast" (Synaptic Homoestasis Hypothesis). The creation of new synapses continues throughout our life, which is why there is (some) neural plasticity even in old adulthood. Thus, the brain itself knows how to get rid of irrelevant information (i.e. whatever it does not use) and is receptive to learning even in old age.

What is more, the brain grows with its tasks. So if you avoid learning new tasks, your brain will effectively decline and so will most cognitive skills (working memory, executive functions, processing speed, reasoning etc.). Likewise, our brain does not easily transfer skills from one task to general cognition. If you start doing Sudokus 5h a day, you will improve your Sudoku skills, but not your general ability to reason, for instance. Thus, to the contrary, it seems that what best maintains cognitive functioning in old age is continued engagement in cognitively challenging activities (see Park 2014) rather than the avoidance of new information.

With multilingualism, there is currently a hot debate about its benefits for cognition. For one, it has been shown that bilinguals have a smaller lexicon (per language) than monolinguals - but generally, vocabulary keeps increasing over the lifespan, so I would not worry about that. Ellen Bialystok is a strong defender of the "bilingual advantage", arguing that cognitive skills are even better preserved in bilinguals than in monolinguals, and she provided cross-sectional data showing that symptoms of dementia are delayed by 4 years in bilinguals as compared to monolinguals. The group of Kenneth Paap challenges these findings, arguing that there is no bilingual advantage see Vaughn 2015 for a discussion, but there is little to no evidence suggesting a disadvantage for cognitive ability in bilinguals. Starting to learn a new language in adulthood is even more promising, as it has been shown to promote grey matter increase in areas also responsible for cognitive functions that are known to decline in old age (see Antoniou et al., 2013).

In summary, I do not know of "information ballast" impairing basic mental mechanisms, but knowing more than one language has certainly not been shown to decrease your mental health.