Measure of the support of a Borel probability on a metric space
Following Pietro's lead, let me observe that if there is a measurable cardinal, then there is a counterexample.
Suppose that $\kappa$ is a measurable cardinal. Then there is a $\kappa$-additive 2-valued measure $\mu$, measuring all subsets of $\kappa$, giving them either measure $0$ or $1$, giving measure $1$ to the whole space and giving measure $0$ to any set of size less than $\kappa$ (among others). If we give $\kappa$ the discrete topology, then every set is closed (and hence Borel), and the support is empty.
Every $\sigma$-smooth measure is $\tau$-smooth. This is what we need. As noted, if there is a (real-valued) measurable cardinal, then this may fail for a metric space. A space is called "measure-compact" iff every $\sigma$-smooth measure is $\tau$-smooth.
The reference for all of this (up to 1965) is: V. S. Varadarajan, "Measures on Topological Spaces". In a completely regular space we would use "zero sets" (a set where some continuous real-valued function vanishes). But in a metric space these are the same as the closed sets. A (finite, Borel) measure $\mu$ on a metric space is $\sigma$-smooth iff it is coutably additive, but this means if $A_n$ is a decreasing sequence of closed sets, then $\mu(A_n)$ converges to $\mu(\bigcap_n A_n)$. A stronger condition on $\mu$ is $\tau$-smooth: if $A_t$ is a decreasing net of closed sets, then $\mu(A_t)$ converges to $\mu(\bigcap_t A_t)$. The "support" of a probability measure $\mu$ is the intersection of all closed sets of measure $1$. And (assuming $\mu$ is $\tau$-smooth) this intersection again has measure $1$.
As I recall, a metric space is measure-compact if and only if there is no discrete subset with real-valued measurable cardinal. So, in particular, if there are no real-valued measurable cardinals, then the answer to the question in the title is YES. Joel has provided the converse. Thus this question is presumably independent of ZFC.
The term "measure-compact" is due to Moran, 1965. By analogy with "real-compact" which may be characterized in the same way using only $\{0,1\}$-valued measures.
Consider an uncountable discrete metric space $X $ (i.e., metrized by the Kronecker delta). Define a measure on $X$ putting for any $A\subset X,\\ $ $\mu(A)=1$ or $\mu(A)=0$ according whether $A$ belongs to a given non-principal ultrafilter $\mathcal{F}$, or not (sigma-additivity holds, for there are no disjoint subsets of positive measure). Then $\mu$ is a Borel probability measure with empty support.
[edit] Actually, this is additive, but to ensure sigma-additivity it would be needed that $\mathcal{F}$ be closed under countable intersections.