"Love and Friendship" by Emily Bronte
"To Meet Old Friends" by Helen Steiner Rice
Rumi, a 12th century mystical Islamic poet has many poems about deep friendship. The friend is referred to as the beloved, but this is not meant to be romantic; rather, it is an expression of deep warmth, friendship, and spirtual sharing.
Is this what you mean--poems about friendship? Or poems that are "romantic" except that they emphasize caring and closeness rather than things like beauty or love?
Tennyson's "In Memoriam" is about death, but also about deep friendship, about caring. Sections like "Doors, where my heart was used to beat...." etc. may fit the bill for you.