I believe that the gang of four themselves classify design patterns as
a common solution to a commonly occurring problem*
So yes, the patterns are relevant when the same type of problem occurs. And this brings us to a problem with the term "Design Pattern". A pattern is something recognizable that occurs repeatedly. So in reality there is not a pattern of designs, there is a pattern of problems.
Some programming languages may have native solutions to some of those problems. The "Design Patterns" book itself mentions that the visitor pattern is of little value if you are using CLOS, as multi-dispatch is natively supported by CLOS, the very problem that the Visitor pattern is trying to solve.
Also, the .NET framework has a build in event mechanism for publishing events to multiple listeners, making the Observer pattern less relevant in this context.
The change from desktop applications to web applications** also change the type of programming problems we have to solve. Many of the patterns in the book "Design Patterns" are relevant for desktop applications, but not so much for web applications. Of course, with single page apps, these patterns may be relevant again on the client-side.
But the design patterns, and books like "Design Patterns", or "Patterns of Enterprise Application Architecture" are of huge value when you are a novice programmer and faced with a new type of problem for the first time; as I was the first time I was asked to implement Undo functionality. Had it not been for the "Design Patterns" book, my implementation would probably had been something like storing a snapshot of the data after each state-changing operation*** - a very error prone, and horribly inefficient, approach.
So yes, some of the pattern become less relevant over time, and as you become an experienced programmer, you think less about them. But to a novice, they are valuable, as long as you remember that they are the means to solve a problem - and not a quest to use as many as possible.
* quote may not be 100% accurate as it is taken from memory
** in my experience, it is getting very common for enterprises to choose web delivery mechanisms for internal line-of-business applications.
*** after learning functional programming and functional data structures, then that might actually be the way I would solve it today.