Do antibiotics help the inflammatory response?
The inflammatory response is actually caused by the immune system when something has been damaged, in order to protect the area of tissue. The damage can be caused by something physical (a burn, a wound) or by some pathogen in an infection.
If the inflammation is caused by a bacterial infection, the antibiotics will eradicate the bacteria, and so eventually the inflammation will go down, no longer being needed if the threat has been killed.
Anti-inflammatory drugs directly target the inflammation. However, since the inflammation is designed by the immune system to help you, you will have to consider whether it is doing more harm than good or not.