No. It is not necessary to merge mate pairs. In fact, it will not always be possible because the read may not overlap. All of the commonly used aligners have an option for processing paired-end data.
Reads may need to be merged for some specialized applications (QIIME, for example), but then that will be explicitly stated.
If your reads do not overlap (most of cases), there is no need to merge them as most sequence aligners have the option for mapping pair-end reads. Otherwise you may wish to perform read merging (occasionally if you perform pair end small RNA seq), please refer to this thread for more details.