Are there any multicellular forms of life which exist without consuming other forms of life in some manner?
Yes, plants for example!

Autotrophs vs. heterotrophs

Plants are autotrophs while animals and fungi are heterotrophs. Have a look at the Wikipedia articles. In short, autotrophs organisms are organisms that synthesize their own compounds from inorganic compounds. Heterotrophs organisms are organisms that synthesize their own compounds from organic compounds.

Therefore, any multicellular plants are multicellular organisms that "exist without consuming other forms of life" (with the exceptions of carnivorous plants).

Consuming nonliving organic matter

Of course, many life forms feed on organic matter but only on non-living organic matter. Such life forms do not kill to feed. It is the case of all detritivores (such as earthworms) and decomposer such as fungi.

Competition vs predation

As @TechZen said, not consuming other organisms doesn't mean that the existence of an individual of this species is not causing the death of another individual (competition and parasitism that does not involve consuming the other individual).