The answer:  Most Likely
Dry areas that don’t get seasonal rain may be able to go without, however, most homes do need gutters.  Gutters prevent damage to your home’s foundation, sidewalks, and interior as they channel thousands of gallons of water off of your roof.
Without gutters you’ll eventually be looking at a soil erosion situation and/or damage to your mulch and garden beds.
Soil erosion can cause your foundation to settle resulting in uneven floors and cracked walls and chimnies.