Tier lists are an important and totally legitimate part of the competitive scene, especially in a game this large. However, the majority of even the competitive community misunderstands or misuses them. Here are the most common misconceptions about tier lists.
1. Tier lists tell you who to play
Tier lists do not make you or anyone else do anything, and they don't change the game at all. Tier lists simply compare the inherent advantages and disadvantages of the characters in the game. The tier list is like a list of observations. Despite what many complainers say, Sheik is top tier because she is good, she is not good because she is top tier. The list is based on highly informed, logical observations of the game, for the most part, but the placements are only as good as our information at the time. Because the lists are just based on current information, they can and do change from time to time. 15 years after Melee's release, aMSa is just now proving the viability of Yoshi, raising him from the lower end of Melee's tier list. Who you play is your own business. Tier lists simply give you a warning of how uphill your battle will probably be.
2. Tier lists apply to all levels of play
The only skill level where tier lists are truly accurate is at the absolute peak of technical skill. The lists are made considering every attack, strategy, and technique a character has available to them. If you cannot perform every attack, strategy, and technique your character has to offer (like 99% of players) then the tier list is not accurate to you. For example, in past games the Ice Climbers were infamous for their desyncs and chain grabs, which put them high on both Brawl and Melee's tier lists. However, if you can't properly chain grab, the Ice Climbers are significantly worse than what the tier list would lead you to believe. Their top tier position is reliant on you being able to use them at the top level of play.
3. Tier lists apply to all modes of play
Tier lists are made assuming a certain ruleset, so they are only accurate for that ruleset. For the competitive Smash tier lists most people talk about, this means 1v1 play with competitive stage selection, no items, and the common stock/time rules used by tourneys. Outside of those very specific conditions, the major tier lists are not accurate. For example, Ganondorf is good in free-for-alls because the chaos of the fight can distract opponents long enough for him to execute brutal smash attacks to net crazy early KO's. In 1v1, though, he very rarely is going to nail anybody with a fully charged fsmash. So in Brawl he was bottom tier, despite his capabilities in 4-player fights. Tier lists are meaningless outside of the ruleset from which they were made.
4. Tier list positions determine match outcomes
Player skill is vastly more important than how good their character is, especially in a game as well-balanced as Smash4. A great player using a bad character will basically always still defeat a bad player using a great character. However, at the pro level most players are very close in skill level. If two pro players of equal skill face off, the one using the better character will have a distinct advantage. Hence why most top-level pro players use good characters; using a bad character could be the difference between victory and defeat!
5. Tier lists compare individual characters
This is probably the one I see most from even good competitive players. A tier list is not a matchup chart. Just because a character is ranked above another does not mean the higher-ranked character has the advantage in that matchup. You need a dedicated matchup chart to make those conclusions. Tier lists compare a character to the cast as a whole. Almost all characters have at least one bad matchup with a character below them, but they simply have more good matchups to make up for it. To cite an extreme example, in Brawl the Ice Climbers were considered to have a bad matchup against Peach, despite Peach being 3 tiers lower than them. The Icies had a huge advantage against almost every other character, though, which is why they were top tier. Now, obviously a character is probably at an advantage over characters far below them, but this data can't be concluded for sure from a tier list.
So after all that talk of what tier lists don't do, what do they do? First off, they give you an idea of which characters you should prepare for when entering the competitive scene. They are not 100% accurate for this, since a character can be both unpopular and good, but the fact that pro players generally use good characters means it's a good place to start. Tier lists also give you an idea which characters will give you the biggest advantage in current tournament play. Tier lists can be very helpful for new players as they choose a character to practice with, and making the lists also promotes meaningful discussion about the advantages and disadvantages of each character.
tl;dr: Yes, tiers matter, but not if you don't take the time to understand them.