Is it possible to consume a library built using C++17 against a C++11 application if all the public facing headers and APIs belonging to the C++17 library that the C++11 application consumes follows C++11 syntax.
In general, that is a recipe for disaster. In a subset of cases it might work if you are lucky (for instance, if you do not share standard library objects that changed ABI, if you do not trigger any ABI difference in your usage of your API, etc.).
What you want to do instead is compile all your code using the exact same compiler, including compiler version and compiler flags. Even then, you should read your compiler's documentation for further possible issues regarding static/dynamic linking of dependencies and system dependencies.
The internal implementation of the C++17 library does have C++17 specific features.
That is not a problem on its own (in fact, many C++ libraries give C interfaces), but you need to respect whatever linking restrictions/issues your compiler/platform documents.
Does it matter if it's a statically linked vs dynamically linked?
Same. For most platforms, it should not change anything with respect the question of mixing C++11 and C++17, but you still have to take care of the usual issues.