SNPs in 3'UTR have been reported to alter the regulational state of proteins by overexpression or downregulation, to estimate that we could retrieve 3'UTR snps related to a specific gene, then asses their expressional state using several tool, like http://utrdb.ba.itb.cnr.it/
it has been also found to alter the miRNA states to inspect that we may use
Expression-wise it could, if it is located near or within regulatory regions.
In regard to the overall function of the protein itself, maybe it´s function could be attenuated or enhanced, but it would be derived from the changes in expression that the SNP could be promoting. From what i´ve learned, it shouldn´t affect the protein´s function. If it is a protein that binds to DNA for example, a SNP at the 3´UTR should not affect its ability to bind to DNA.