Instead of relying on human-annotated training samples to build a classifier, weakly supervised scientific paper classification aims to classify papers only using category descriptions (e.g., category names, category-indicative keywords). Existing studies on weakly supervised paper classification are less concerned with two challenges: (1) Papers should be classified into not only coarse-grained research topics but also fine-grained themes, and potentially into multiple themes, given a large and fine-grained label space; and (2) full text should be utilized to complement the paper title and abstract for classification. Moreover, instead of viewing the entire paper as a long linear sequence, one should exploit the structural information such as citation links across papers and the hierarchy of sections and paragraphs in each paper. To tackle these challenges, in this study, we propose FUTEX, a framework that uses the cross-paper network structure and the in-paper hierarchy structure to classify full-text scientific papers under weak supervision. A network-aware contrastive fine-tuning module and a hierarchy-aware aggregation module are designed to leverage the two types of structural signals, respectively. Experiments on two benchmark datasets demonstrate that FUTEX significantly outperforms competitive baselines and is on par with fully supervised classifiers that use 1,000 to 60,000 ground-truth training samples.