Most of the scholarly work on poetry is published either as journal articles or as single chapters of edited books which have a non-poetry focus. However, the volumes below are full-length studies devoted to poetry.
Hutchinson, Peter, ed. Landmarks in German Poetry. Lang, 2000. [examines a selection of famous, single poems] Link
Prawer, Siegbert. Heine: Buch der Lieder. Arnold, 1960. [old and small but still the best introduction to Heine]
Ryan, Judith. Rilke, Modernism and Poetic Tradition. CUP, 1999. [does what the title says]
Cosentino, Christine. Tierbilder in der Lyrik des Expressionismus. Bouvier, 1972. [unusual angle]
Arnold, Heinz Ludwig, ed. Lyrik des 20. Jahrhunderts. Text & Kritik, 1999. [not as comprehensive as the title promises but some interesting essays]
Kuhn, Tom, and Karen Leeder, eds. Empedocles' Shoe: Essays on Brecht's Poetry. Methuen, 2002. [a mix of essays, some on single poems] Link
Wenzel White, Erdmute. The Magic Bishop: Hugo Ball, Dada Poet. Camden House, 1998. [interesting and original] Link
Bridgwater, Patrick. The German Poets of the First World War. London, 1985. [good topic, nicely done]
Bower, Kathrin M. Ethics and Remembrance in the Poetry of Nelly Sachs and Rose Ausländer. Camden House, 2000. [important poets] Link
Divers, Gregory. The Image and Influence of America in German Poetry since 1945. Camden House, 2002. [interesting concept] Link
Berendse, Gerrit-Jan. Die sächsische Dichterschule: Lyrik in der DDR in den sechziger und siebziger Jahre. Lang, 1990. [introduces a generation]
Leeder, Karen. Breaking Boundaries: A New Generation of Poets in the GDR. OUP, 1996. [definitive on 1980s poetry] Link
I am still waiting to read a really good book on Celan’s poetry, so none is listed here.
See also The Poet's Role