The problems you’d see from failing to tell the compiler about registers you modify would be exactly the same as if you wrote a function in asm that modified some call-preserved registers1. See more explanation and a partial example in Why should certain registers be saved? What could go wrong if not?
In GNU inline-asm, all registers are assumed preserved, except for ones the compiler picks for "=r" / "+r" or other output operands. The compiler might be keeping a loop counter in any register, or anything else that it’s going to read later and expect it to still have the value it put there before the instructions from the asm template. (With optimization disabled, the compiler won’t keep variables in registers across statements, but it will when you use -O1 or higher.)
Same for all memory except for locations that are part of an "=m" or "+m" memory output operand. (Unless you use a "memory" clobber.) See How can I indicate that the memory *pointed* to by an inline ASM argument may be used? for more details.
Footnote 1:
Unlike for a function, you should not save/restore any registers with your own instructions inside the asm template. Just tell the compiler about it so it can save/restore at the start/end of the whole function after inlining, and avoid having any values it needs in them. In fact, in ABIs with a red-zone (like x86-64 System V) using push/pop inside the asm would be destructive: Using base pointer register in C++ inline asm
The design philosophy of GNU C inline asm is that it uses the same syntax as the compiler internal machine-description files. The standard use-case is for wrapping a single instruction, which is why you need early-clobber declarations if the asm code in the template string doesn’t read all its inputs before it writes some registers.
The template is a black box to the compiler; it’s up to you to accurately describe it to the optimizing compiler. Any mistake is effectively undefined behaviour, and leaves room for the compiler to mess up other variables in the surrounding code, potentially even in functions that call this one if you modify a call-preserved register that the compiler wasn’t otherwise using.
That makes it impossible to verify correctness just by testing. You can’t distinguish “correct” from “happens to work with this surrounding code and set of compiler options”. This is one reason why you should avoid inline asm unless the benefits outweigh the downsides and risk of bugs. https://gcc.gnu.org/wiki/DontUseInlineAsm
GCC just does a string substitution into the template string, very much like printf, and sends the whole result (including the compiler-generated instructions for the pure C code) to the assembler as a single file. Have a look on https://godbolt.org/ sometime; even if you have invalid instructions in the inline asm, the compiler itself doesn’t notice. Only when you actually assemble will there be a problem. (“binary” mode on the compiler-explorer site.)
See also https://stackoverflow.com/tags/inline-assembly/info for more links to guides.