Current and reversible heat in battery reaction
How is the current related to nF? current and nF have different dimensions?


$nF$ is the charge $C$ transferred during the reaction, while current $i$ is the rate of charge transfer ($dC/dt$).

The following is a derivation, now edited to be more rigorous. The equation you provide is an expression for electrical Joule heating, which follows in the steady-state from the expression for the power (rate of work) generated by the battery. From Ohm's law 

$P=\left(\frac{dw_{rev}}{dt}\right)=-i^2R=-i E_0 \tag{1}$

where $E_0$ is the (open-circuit) electric potential, $i$ the current, $w_{rev}$ is the reversible electrical work. Note that $E_0$ is the potential when reversible work is done.

Now assume a steady state, with constant P and T constraints, with work done and heat generated cancelling (so that the internal energy $U = constant$), that is

$Q_{rev} = -w_{rev} \tag{2}$

and therefore also power and rate of heat dissipation equal:

$\left(\frac{dQ_{rev}}{dt}\right)=-\left(\frac{dw_{rev}}{dt}\right)=i E_0 \tag{3}$

Now according to the Nernst equation, 

$\Delta G=w_{rev} =-nFE_0 \tag{4}$

In the steady-state the dissipated heat is equal to the electrical work, so that, combining (2) and (4), we have that 

$\frac{Q_{rev}}{nF}=-\frac{w_{rev}}{nF}=E_0 \tag{5}$

which leads, combined with (3), to the equation in the book:

$\left(\frac{dQ_{rev}}{dt}\right)=i \frac{Q_{rev}}{nF} \tag{6}$



Aside 

Note I apply the opposite sign convention for work from that in the OP (apologies) - in the convention I use work is positive when performed on the system (charging a battery is positive work, discharging negative work). This does not affect the result of the derivation, since I apply the same sign convention for heat (for an exothermal process heat is negative).



On $\Delta H$ and an alternative (longer) derivation

For a process at constant p, it is common to encounter the expression

$$\Delta H = Q_p$$

However if there is non-PV work, the more general form of this equation is  

$$\Delta H = Q_p + w_{non-pV}$$

This expression is general and applies when the process is carried out either reversibly or irreversibly. In differential form,

$$d\Delta H = dQ_p + dw_{non-pV} \tag{a1}$$

Note also that in the preceding derivation non-pV work is electrical. 

When the process is carried out reversibly a maximum amount of work is done and 

$$w_{non-pV,rev} =\Delta G = \Delta H - T \Delta S = \Delta H - Q_{rev}$$

This leads to the following expression at constant T, equal to that in the linked problem (but note the different work sign convention here):

$$ d\Delta H = dQ_{rev} + dw_{rev} = d\Delta G + Td\Delta S \tag{a2}$$

where I dropped the "non-pV" subscript since the work is assumed to be electrical.

Since H is a state function it must be equal for reversible and irreversible processes, and we can equate (a1) and (a2), leading to the following general expression:

$$ d\Delta H = dQ_p + dw_{ele}=  d\Delta G + Td\Delta S $$

which leads finally to 

$$ dQ_p = - dw_{ele} +  d\Delta G + Td\Delta S  \tag{a3}$$

Taking the time derivative of this equation results in the final expression in the linked problem (making sure to account for differences in the work sign convention):

$$\dot{Q} = \dot{Q}_\text{rev} + \dot{Q}_\text{irrev} = IT\,\frac{\mathrm dE_0}{\mathrm dT} + I(E-E_0) \tag{a4}$$

To obtain the equation in this problem it is only necessary to apply the reversibility condition starting from either (a3) or (a4). Starting from (a3), 

$$ dQ_{rev} = - dw_{ele,rev} +  d\Delta G + Td\Delta S  $$

But $ dw_{ele,rev} =  d\Delta G  $ which leads to (somewhat trivially) 

$$ dQ_{rev} = Td\Delta S  $$

Taking the time derivative and inserting the Nernst expression for $\Delta S$ (twice!)  gives

$$\dot{Q}_\text{rev} = \left(\frac{dQ_{rev}}{dt}\right) = IT\,\frac{\mathrm dE_0}{\mathrm dT} = I\frac{Q_{rev}}{nF}$$

which is the desired expression. The same result can be arrived at by applying the reversibility condition to equation (a4).