The persistent discrepancy between local determinations of the Hubble constant $H_0$ and the Planck 2018 value ($67.4 \pm 0.5~{\rm km\,s^{-1}\,Mpc^{-1}}$) within $\Lambda$CDM remains a central challenge in precision cosmology. We investigate the Hubble tension in $\Lambda$CDM and its Rastall extension (R-$\Lambda$CDM) for flat, open, and closed geometries. We analyze three primary dataset combinations: D$_1$ (late-time probes: SN + $H(z)$ + $f\sigma_8$), D$_2$ (late-time probes combined with DESI DR2 BAO and BBN), and D$_3$ (late-time probes combined with BAO and Planck 2018 CMB distance priors). Parameters are constrained via Markov Chain Monte Carlo sampling, and tensions with SH0ES ($73.2 \pm 1.3~{\rm km\,s^{-1}\,Mpc^{-1}}$) and Planck are expressed in units of the combined uncertainty. In addition, we include a Planck-only configuration (D$_4$) as a reference baseline to isolate early-Universe constraints on $H_0$. Within $\Lambda$CDM, D$_1$ and D$_2$ yield $H_0 \simeq 70.75$-$71.43~{\rm km\,s^{-1}\,Mpc^{-1}}$, reducing the SH0ES discrepancy to $1.11\sigma$--$1.63\sigma$ while maintaining a $3.62\sigma$-$4.23\sigma$ tension with Planck. Including CMB distance priors (D$_3$) shifts the result to $H_0 \simeq 67.18$--$67.55~{\rm km\,s^{-1}\,Mpc^{-1}}$, consistent with Planck at $0.09\sigma$-$0.38\sigma$ but increasing the SH0ES discrepancy to $4.25\sigma$-$4.51\sigma$. A similar dataset-driven behavior is found in R-$\Lambda$CDM. For D$_1$ and D$_2$, we find $H_0 \simeq 70.79$--$71.48~{\rm km\,s^{-1}\,Mpc^{-1}}$, with SH0ES tensions of $1.11\sigma$-$1.60\sigma$ and Planck discrepancies of $3.64\sigma$-$4.19\sigma$. For D$_3$, the inferred value becomes $H_0 \simeq 68.50$--$69.50~{\rm km\,s^{-1}\,Mpc^{-1}}$, reducing the Planck tension to $1.70\sigma$-$2.95\sigma$ and yielding a SH0ES discrepancy of $2.65\sigma$-$3.45\sigma$. In this configuration, $H_0$ lies between the Planck and SH0ES determinations, partially alleviating their discrepancy. This behavior can be interpreted as a consequence of the modified matter evolution in Rastall cosmology, which induces a degeneracy between $\epsilon$ and $H_0$ at the background level. Small positive values $\epsilon \sim \mathcal{O}(10^{-3})$, favored by CMB-inclusive datasets, lead to a mild but systematic increase in $H_0$ relative to $\Lambda$CDM, while the effect vanishes as $\epsilon \to 0$. Model comparison using AIC and BIC shows statistical equivalence between $\Lambda$CDM and R-$\Lambda$CDM for D$_1$ and D$_2$, while in D$_3$ the R-$\Lambda$CDM extension is preferred in flat and closed geometries and decisively favored in the open case. Overall, the results highlight that any apparent alleviation of the Hubble tension remains strongly dependent on dataset composition and does not provide a universal resolution within the considered framework.
