Recent advances in large language models have facilitated the development of unified speech language models (SLMs) capable of supporting multiple speech tasks within a shared architecture. However, tasks such as automatic speech recognition (ASR) and speech emotion recognition (SER) rely on distinct types of information: ASR primarily depends on linguistic content, whereas SER requires the integration of both linguistic and paralinguistic cues. Existing multitask SLMs typically adopt naive parameter sharing or prompt-based conditioning without explicitly modeling the differences in information composition required by each task. Such designs risk task interference and performance degradation, especially under limited data conditions. To address these limitations, we propose HarmoniFuse, a component-selective and prompt-adaptive framework for multi-task speech language modeling. HarmoniFuse is designed to harmonize heterogeneous task demands by selecting and fusing task-relevant components of speech representations. Specifically, it integrates a gated speech encoder to extract task-specific acoustic features and a prompt-adaptive dynamic fusion module to aggregate transformer layers based on task characteristics. In addition, a batch-interleaved training strategy enables leveraging separate ASR and SER datasets without requiring joint annotation. Experimental results demonstrate that HarmoniFuse improves both ASR and SER performance, offering a scalable and robust solution for multitask speech understanding under realistic data constraints.