As you haven't said what you mean by "modern" I will make up my own definition and then answer that question. Let's say that a primitive LMS is a collection of tools--for resource sharing, discussion, syllabus, assignments, quiz/test, etc. The primitive LMS was largely used for pushing information and course content to students. In blended or online classes the discussion forum was also used for discussion activities. The primitive LMS was about getting students to jump through the right hoops, in FERPA-induced privacy.
A modern LMS (if one yet exists) is task-oriented rather than tool-oriented. Instead of presenting the user with a list of tools as the primary structural element, it instead presents users with a course. Thus the schedule, whether time-based or a list of modules/units or assignments is the primary structural element. The tool aspect is tightly integrated into the module approach such that I do not have to navigate up and down hierarchical tools in order to move from a reading to a discussion--I can directly link or even annotate socially within the reading.
So that is my first and most fundamental feature of the modern LMS: task orientation. The architecture should fit with the activity rather than the activity having to be unnaturally disassembled and reconstructed to fit the convenience of the programmers.
But here's my list:
task-oriented - (as described above) the architecture is based on modules/units/weeks, with tools accessible at the point of need
open participation - openness is an option rather than the LMS being a locked garden that no one outside the class can ever see and to which you lose the key when you graduate; participation can be extended beyond the traditional boundaries of enrolled students and faculty of record; guest participants or public (MOOC) participants can easily be involved; current students can learn from and form relationships with alumni
scalable - can handle hundreds or thousands of participants without participants getting lost in the crowd; for instance, students can figure out how to form study groups or other kinds of subgroups rather than trying to interact meaningfully beyond the scale of human capability
competency-aware - there are was to demonstrate pre-existing competencies or masteries so as to not waste time pretending to learn when the knowledge or skills are already present
serendipitous - support for informal, ad hoc affiliations and activities
status-aware - status of progress, deadlines to meet, opportunities to know about, etc., are clearly visible and you don't have to hunt for them up and down a deep tool hierarchy
identity-aware - education is all about constructing a personal trajectory; but the construction is social; I need a way to present my academic self and interact with other selves similarly presented, whether those of instructors or students
formative - learning is gradual and iterative; the iterations are important, both to the reflective learner and the assessing teacher; intermediate states should be save and be viewable; the course itself can be in continuous formation, with students as well as instructors being able to suggest new content, new activities, new purposes, new directions; the LMS needs to be flexible to support such changes of direction
So I haven't provided a list of tools. But when I think about what an online learning environment (in contrast to an industrial-age learning management system) should provide, these are the sorts of things that come to mind.