Signal peptide is located at the N-terminal of the protein. It is cleaved off after sorting through the membranes of the organelles in the secretory system like the endoplasmic reticulum, golgi apparatus lysosome.
Transit peptide is more commonly found at the C-terminal end and it targets protein to microbodies which are single membrane-bound small organelles such as chloroplast, mitochondria, peroxisome.
I feel mutations in transit peptide may or may not affect protein function because the function of transit peptide is to transport the protein for further processing or action. Because of mutation if transit peptide fails to perform its function then the function of the protein could be affected.
First of all, the term signal peptide and transit peptide are often (though strictly speaking incorrectly) used interchangeably. For a stricter use of these terms see for example table 1 in this paper:
Kunze, M., & Berger, J. (2015). The similarity between N-terminal targeting signals for protein import into different organelles and its evolutionary relevance. Frontiers in Physiology, 6, 259.
Indeed, there are indications that these targeting peptides might have additional functions. There are studies that show effects on for example the folding of (the mature part of) the protein. See for example: Singh, P., Sharma, L., Kulothungan, S. R., Adkar, B. V., Prajapati, R. S., Ali, P. S. S., ... & Varadarajan, R. (2013). Effect of signal peptide on stability and folding of Escherichia coli thioredoxin. PloS one, 8(5), e63442.
It depends on what you mean by affecting the function. Indeed, as indicated by Malcolm Nobre a mutation that leads to a less functional signal or transit peptide will affect the targeting of your protein and (indirectly) affect its functioning. The mutation can affect the binding to proteinaceous components involved in the secretion machinery:
Fekkes, P., De Wit, J. G., Van Der Wolk, J. P., Kimsey, H. H., Kumamoto, C. A., & Driessen, A. J. (1998). Preprotein transfer to the Escherichia coli translocase requires the co‐operative binding of SecB and the signal sequence to SecA. Molecular microbiology, 29(5), 1179-1190.
Or influence the ability of a signal peptide to interact with phospholipids:
Article Anionic phospholipids are essential for α-helix formation of...
Article Tryptophan fluorescence study on the interaction of the sign...
A signal peptide (sometimes referred to as signal sequence, targeting signal, localization signal, localization sequence, transit peptide, leader sequence or leader peptide) is a short peptide (usually 16-30 amino acids long)