Problems with quotes

Polishing - Because we cannot change the text, quotes are not subject to our normal iterative editing process (iterative meaning multiple people polishing it over time)
WP:UNDUE weight issues. We're giving a specific person a platform to say what they want in a lot of detail, instead of summarizing them.
WP:PRIMARY issues. Wikipedia is supposed to be a WP:TERTIARY source. We're supposed to concisely summarize knowledge from secondary sources. Quotes are often primary sources.
WP:POV - Quotes are often a vehicle for POV. POV that would normally be polished out of non-quotes will often be left in quotes, because we can't change the text in quotes.
Clarity - Quotes are not usually as clear and concise as summary style prose. Unclear text increases mental burden on the reader.
Copyright - Arguably, long quotes are a violation of copyright, unless they are in the public domain. See WP:COPYQUOTE, which states that we shouldn't use long quotes when a short quote will suffice. Also see Wikipedia:Copying text from other sources#What about quotes?, which states that quotes of copyrighted text should not be used if we can easily replace the quote with free text that we write ourselves.

Specific types of quote sections

Reaction sections - Articles based on a top news story often end up with large "Reactions" sections, where every country that makes a statement is quoted, and a citation to a WP:PRIMARY source such as Twitter is used. These sections often become larger than the entire rest of the article, and are a prime example of why quotes and primary sources are bad. See WP:REACTIONS, WP:RSPTWITTER, WP:UNDUE. Best to wait for secondary sources to cover this (they figure out how important each country's statement is for us), and then to summarize, grouping by each country's position. (Countries A, B, and C condemned the action, and countries D, E, and F supported the action.)
Views sections - Sometimes an article will have a "views" section that, while in paragraph form, quotes many people such as experts and politicians. These sections are better re-written without the quotes, summarizing the key arguments for and against. (Supporters of X argued A, B, and C. Opponents of X argued D, E, and F.)

On the other hand, there are some rare situations where quotes are quite useful.

To quote a dictionary definition. Often, trying to paraphrase a dictionary definition will significantly alter the meaning.

Recommend never using this by default. It makes the article's reference section huge.
- Examples (don't do this): [1][2]
It is also a red flag for original research. If you have to quote from the article, you may be making an argument that is too complex. Remember, sources need to directly state their conclusions, and we as encyclopedists cite and summarize those conclusions.
It also makes reusing the same citation harder. For example, if you want to use something else said in that citation, you would in theory need a different citation with a different quote.
Only use the quote parameter of cite templates if someone adds a ((Failed verification)) template to your citation, meaning they read your citation and could not find the section of it that supports the claim made. In that case, add a quote for that one citation only.