The practical utility of Speech Emotion Recognition (SER) systems is undermined by their fragility to domain shifts, such as speaker variability, the distinction between acted and naturalistic emotions, and cross-corpus variations. While domain adaptation and fine-tuning are widely studied, they require either source data or labelled target data, which are often unavailable or raise privacy concerns in SER. Test-time adaptation (TTA) bridges this gap by adapting models at inference using only unlabeled target data. Yet, having been predominantly designed for image classification and speech recognition, the efficacy of TTA for mitigating the unique domain shifts in SER has not been investigated. In this paper, we present the first systematic evaluation and comparison covering 11 TTA methods across three representative SER tasks. The results indicate that backpropagation-free TTA methods are the most promising. Conversely, entropy minimization and pseudo-labeling generally fail, as their core assumption of a single, confident ground-truth label is incompatible with the inherent ambiguity of emotional expression. Further, no single method universally excels, and its effectiveness is highly dependent on the distributional shifts and tasks.