Clinical variant classification of pathogenic versus benign genetic variants remains a challenge in clinical genetics. Recently, the proposition of genomic foundation models has improved the generic variant effect prediction (VEP) accuracy via weakly-supervised or unsupervised training. However, these VEPs are not disease-specific, limiting their adaptation at the point of care. To address this problem, we propose DYNA: Disease-specificity fine-tuning via a Siamese neural network broadly applicable to all genomic foundation models for more effective variant effect predictions in disease-specific contexts. We evaluate DYNA in two distinct disease-relevant tasks. For coding VEPs, we focus on various cardiovascular diseases, where gene-disease relationships of loss-of-function vs. gain-of-function dictate disease-specific VEP. For non-coding VEPs, we apply DYNA to an essential post-transcriptional regulatory axis of RNA splicing, the most common non-coding pathogenic mechanism in established clinical VEP guidelines. In both cases, DYNA fine-tunes various pre-trained genomic foundation models on small, rare variant sets. The DYNA fine-tuned models show superior performance in the held-out rare variant testing set and are further replicated in large, clinically-relevant variant annotations in ClinVAR. Thus, DYNA offers a potent disease-specific variant effect prediction method, excelling in intra-gene generalization and generalization to unseen genetic variants, making it particularly valuable for disease associations and clinical applicability.