I don’t believe that this question can simply be answered with a “yes” or “no”. It is possible that in some cases it does not really matter if what we believe is actually true, but in other cases it is greatly important to distinguish between real and false.
In some sciences it is very important that what we believe is true because for example if we study biology and later on become a doctor and are wrong about how to treat a patient we might harm them or maybe even lead to their death. So in such areas it is very important that what we learn and believe is true. What is interesting though is that sciences are mostly based on theories. For example discoveries in physics are mostly based on the belief that atoms exist and have a certain structure, everything seems to be explained by this belief, but in the end atoms and their structure and their smaller particles are impossible to see, so it is unclear if this theory is actually true. In this case we can say that it does not really matter if what we believe is true because even though atoms might be something entirely different in reality we can still apply the current theory in real life and explain most laws of nature by it. So even though this belief might be false we are still able to rely on the further findings to make life easier and basically understandable. So even in sciences it does not always matter if what we believe is true.
Something far different from science is religion and in the case of religion it doesn’t matter if what we believe is true or not. This is because for most people religion is simply something to help them throughout their life. It is like an invisible friend that helps people through all kinds of situations. It encourages people, helps them find a purpose, helps them through bad times, causes people to have faith and simply believe in something bigger then themselves. Of course sometimes it does even matter if our beliefs are true when it comes to religion. For example if someone believes in very extreme things such as thinking that they could survive anything because they believe that their god won’t let them die, this will most probably not have a happy ending. So, one should definitely still be reasonable while believing in a certain greater power.