The simple answer is yes.
Let's look at some data. An analysis released a couple of years ago by the National Education Policy Center uses an econometric take on the issue, where they try to disentangle data to find causal links. In their work, share evidence that class size increases harms students. This is from data that is "econometric" friendly, where the savings today in increasing class sizes (K-12) is offset in the future by far greater costs. (Penny wise, pound foolish?)
The point I want to get to is a very simple one. Another perspective of this issue is as a classroom teacher. Let's look at the sequence of class sizes to see how teaching decisions are affected as class size increases. (The assumptions here include assuming we are talking about the college math setting.) Also I'm not going to list all possibilities. The goal here is to see a pattern as class size increases.
Class size of 10: Anything can be done at this class size. Projects, full IBL (where students present proofs or solutions), team/group work, seminar or whatever comes to mind.
Class size of 20: At 20, things are still manageable, and an instructor can get to know all of the students, customize materials and learning experiences, projects are still doable. IBL is doable in with small groups/pairs and students presenting individually.
Class size of 40: Things start to get to a point where there are too many students to let them present their work to the whole class. Each student may only go to the board a couple of times a term. Somewhere between 30 and 40, instructors tend to switch away from some student-centered methods (such as students presenting their ideas at the board). Small groups can still be used, and the instructor may not be able to get to all groups on a particular task.
Class size of 80: Individual student presentation of mathematics is almost surely off the table. Small groups and peer instruction remain, so instructors can develop this area to support an active learning environment. Projects are highly difficult to implement at this class size, especially at institutions without teaching assistants. The instructor can still visit with some groups and make it around the room (depending on the space) say one trip per class period (approximately).
Class size of 160: At this point, you are well into large lecture territory. Peer instruction (Think-Pair-Share) and similar methods (with or without clickers) remain in play, although the size or difficulty of the task may be on the less challenging side of the spectrum. Presentations by students, class discussions, projects, instructor visits to each group regularly are almost surely off the table.
As you go up in class size, you lose the implementability of teaching strategies that engage students and also anything that does get implemented needs to be done at a higher skill level and attendant time in preparing for class. Managing a discussion with 160 students takes skill, and most all instructors may be inclined to avoid it.
Consequently, what we are seeing is that there exists research evidence suggesting that a negative, material impact is a consequence of increasing class size AND from a practitioner standpoint instructors have fewer, high-impact strategies at their disposal as classes get larger. We have said nothing yet about how class dynamics can change as you get to larger class sizes. For example, it's easier to hide in a large class, and it's easier for students to checkout, be distracted, and show up late.
Another facet of class size issues is that class size is often a policy decision, not in direct control by an instructor. Instructors can have input on class size, but it's not up to instructors to set class size limits. This is done by policy makers or administrators, yet these decisions have significant impact on teaching decisions and hence learning outcomes. Faculty and administrators should take into account the big picture when it comes to class size. Focus on things like nominal efficiency, should be viewed while also heavily weighing learning outcomes, DFW rates and long-term impacts on student learning.
