Not unless you are using features that the target platform does not support.
If you want to use Effect files from a previous XNA project or an XNA sample, you'll need to process them with the MonoGame Effect processor to compile them for that specific platform. Some of these use OpenGL rather than DirectX as their graphics API, so the Effect file from XNA will need to be converted to OpenGL shader language for it to work.
Rather than have developers rewrite their shaders to GLSL, MonoGame installs some tooling to automatically convert the HLSL in the Effect file to the appropriate shader language for the target platform. For the DirectX-based platforms, MonoGame uses the DirectX 11 tool chain to compile your Effect into a shader optimized for that platform. For the OpenGL-based platforms, the Effect is processed by a tool called MojoShader, which does a low-level conversion from HLSL to GLSL that allows the Effect to work on that platform.
Of course, this conversion process can occasionally introduce errors or unsupported features that the target platform does not support. For example, with OpenGL Shader Model 3.0, you cannot do a texture lookup in a Vertex Shader, so if your Effect uses that feature it probably won't work.
Source: From XNA to MonoGame