The homogenous Maxwell equations
$$ \partial_\mu \tilde{F}^{\mu \nu} =0 $$
follow "trivially" from the definition of the dual field strength tensor $\tilde{F}^{\mu \nu} = \epsilon^{\mu \nu \sigma \rho} F_{\sigma \rho}. $
They express a conserved current equation for each component $\nu$. For example, for $\nu=1$, we have
$$ \partial_\mu \tilde{F}^{\mu 1} =0 \quad \leftrightarrow \quad \partial_\mu j^\mu =0 ,$$
where I have defined $j^\mu \equiv \tilde{F}^{\mu 1} $.
When we have a conserved current, we have a conserved charge thanks to Gauss' theorem:
$$ Q \equiv \int_V dx^3 j^0 \quad \text{ with} \quad \partial_0 Q =0 .$$
What is this conserved charge?
It doesn't follow from Noether's theorem and is always conserved irrespectively of if the equations of motion are fulfilled or not, i.e. even "off-shell".
In addition, we get one such conserved charge for each component $\nu$ as mentioned above, so there are, in total, 4 conserved charges here.
In terms of the gauge potential $A_\mu$, the conserved charge reads:
$$ Q= \int_V dx^3 j^0 = \int_V dx^3 \tilde{F}^{0 1} = \int_V dx^3 \epsilon^{01 \sigma \rho} F_{\sigma \rho} = \int_V dx^3 \epsilon^{01 \sigma \rho} ( \partial_\sigma A_\rho - \partial_\rho A_\sigma ).$$
The usual electrical charge associated with the electromagnetic field is
$$ Q= \int_V dx^3 j^0 \quad \text{ where } \quad j^\mu = \partial_\nu F^{\nu\mu}.$$ $$ \rightarrow Q= \int_V dx^3 \partial_\nu F^{\nu0} = \int_V dx^3 \partial_nu ( \partial_\nu A_0 - \partial_0 A_\nu ). $$
