Does bacteria infection cause damage to body tissues by also competing with body tissues for resources and nutrients?

Indirectly: This would be an indirect competition for the body's nutrients as it combats the damage done by bacteria and marshals resources to kill the bacteria which are invading tissues in the body. That is why the old adage of starving a fever is bad advice during a bacterial infection. You want to maintain a good nutritional status as you combat the infection.

Not really: the primary cause of body tissue destruction in context of most severe infections relates to the immune response to the insult. Phase I involves destruction of infected tissue and this is followed by anti-inflammatory properties and wound healing once the insult is neutralized and inflammatory signals fall away.