Drupal has been around for almost 15 years now, I personally have prototyped and designed websites integrated with Drupal for 4. I would be lying if I said Drupal or any CMS has no impact on the user experience considerations of a website, but in most cases a CMS should not. Due to the ever expanding nature of Drupal and the open source module community, it is easy to implement anything that a client or we may require from the CMS. While not directly impacting what we may create and scope out at the early stages of a project, there are some small elements to consider especially when scoping out a project.
If the modules used are hard to manipulate, this can impact what we may be able to collect from a user, especially on tighter budgets. An example of this would be the ability to add an email signup form. In most cases we would generally design up the collection of name and email address, keeping it quick and engaging for the user, but if you use a module which can only collect first name, last name, and email separately this could greatly impact the space that was designed up for your email sign up form. To get around this problem, we would always scope out the requirements of such an adition to make sure the designed element reflects what we can collect, and ultimately what the client requires.
Clear, engaging design
A core goal of any project is to present information in a clear and engaging manner. If an approach is too busy, and overwhelming the user will ultimately feel lost, and go elsewhere to get what they were after. At the same time, the page of content that requires editing could also be daunting to whoever is managing the site. While this is not a localised problem to Drupal, it is worth considering how the content you are designing will look to the content editor, and how they will manage it.
Content structure & SEO
Drupal has the perfect SEO feature set built, when prototyping up a website we have to consider the hierarchy of content on a page, title structures etc. Knowing that drupal does this pretty much out of the box, allows us to be more progressive when it comes to the page structure, and still create the organic search results you may require.
These are not huge problems to overcome, and are what most people come to expect from a modern CMS, which is a testament to Drupal, and its ability to not get in the way of the goals of a new project and its users.