Yes, it can—but the damage is not automatic. A concrete patio in full sun can become extremely hot, often hot enough to feel painful through shoes. That heat can move into an urban planter box in two ways: by conduction through the base and lower sides, and by radiant heat from the surrounding concrete. If the planter is thin-walled, dark, metal, small, or sitting directly on the concrete, the soil near the bottom and edges can get much warmer than the air temperature. Roots are less tolerant of heat than many gardeners expect. When root-zone soil stays above about 90°F (32°C), root function begins to decline; if it climbs toward 100°F (38°C) or higher for hours, fine roots can be injured or killed. You may see wilting at midday even when the soil is moist, yellowing leaves, slow growth, crispy or brown roots, and a general decline that looks like under-watering or root rot. The top of the soil may look fine while the bottom is overheated, so do not judge only by surface dryness.
That said, a well-built planter with thick wood, insulation, and a large soil volume can buffer a lot of heat. The key is to keep the planter from sitting in direct contact with scorching concrete and to keep the root zone shaded, insulated, and evenly moist. Raise the box 1–2 inches off the patio using sturdy feet, pavers, or wooden slats so air can flow underneath. Insulate the base and sun-facing sides with rigid foam board, reflective bubble wrap, wood, or a second pot. Light-colored containers and reflective wraps help, and afternoon shade cloth over the lower sides can reduce radiant heat. Add 2–3 inches of mulch to keep the surface cool and retain moisture, but keep it away from stems. Water deeply in the morning, check moisture at root depth, and avoid shallow daily watering that leaves the lower root zone dry or overheated. A cheap soil thermometer pushed 3–4 inches into the soil is the best way to know; if afternoon readings stay above 90°F, take action. If roots have already been damaged, move or shade the planter, improve insulation, trim dead roots, and repot into a well-draining mix. With a few heat barriers, an urban planter box can sit on a hot concrete patio without cooking its roots.