For example, are there certain things when writing an operating system that cannot be accomplished in a turing complete language?
No. or at least if you found one that would be a disproof of the Church Turing Thesis.
However, there are languages that are Turing complete but in which it is a complete pain in the ass to write certain programs, i.e., string processing in FORTRAN, numerical programming in COBOL, integer arithmetic in sed, practically everything in x86 assembler.
And then of course there's brainfuck.