What you are describing is a giant urban heat island (UHI).
I'm oversimplifying it here, but basically concrete and asphalt absorb more heat and release it slower than rural areas and this effect can be exaggerated where there are tall buildings and narrow streets as this traps the air and therefore the heat.
The difference in temperature between urban and rural areas is usually greatest in the evening as the cooling process is slowed in urban areas.
This warming effect isn't known to be exaggerated over time, i.e. cities don't continuously get warmer.
City sizes can affect the evening cooling process, so the center of an area with tall buildings and narrow streets will cool after the outskirts of that area. But they won't necessarily reach a higher temperature in the first place just because they are the center of this urbanised area.
So in general, your planet sized city is likely to be a few degrees warmer than it would be if left rural, especially at night.
UHIs in hotter climates are known to effect rainfall, as clouds will rise over these warm air pockets leading to precipitation. So more rain around the coast of you giant UHI, this will especially be true the closer you get to the equator and areas where the prevalent wind comes from the ocean. Of course, with no rural land, that is a lot of water to drain away. Precipitation will still be affected by natural geography to a point though (e.g. how far above sea level an area is).