So my question is if I estimate a VAR in first-differences, is there any problem with just applying the forecasted first-differences iteratively with an initial value equal to the last observed data point in the time-series?

Yes, for an $h$-step-ahead forecast you may simply take the $h$ predicted differences for time periods $t+1,\dots,t+h$ and add them to the last observed value $y_t$ to obtain the forecast $\hat y_{t+h|t}$ for $y_{t+h}$ as of time $t$: $$ \hat y_{t+h|t} = y_t + \widehat{\Delta y}_{t+1|t} + \dotsc + \widehat{\Delta y}_{t+h|t} $$

Is this equivalent to taking the first-differenced VAR model (i.e. the coefficients) and applying it to the level data to obtain forecasted levels?

No. Here is a counterexample: consider an estimated model in first differences, $$ \begin{pmatrix} \Delta y_{1,t} \\ \Delta y_{2,t} \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} \begin{pmatrix} \Delta y_{1,t-1} \\ \Delta y_{2,t-1} \\ \end{pmatrix} + \begin{pmatrix} v_{1,t} \\ v_{2,t} \\ \end{pmatrix}. $$ It will produce forecasts $\widehat{\Delta y}_{i,t+h|t}=h\Delta y_{i,t}$ and thus $\hat y_{i,t+h|t}=y_{i,t}+h\Delta y_{i,t}$ for $i=1,2$.
However, when the estimated coefficients are applied on data in levels, $$ \begin{pmatrix} y_{1,t} \\ y_{2,t} \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} \begin{pmatrix} y_{1,t-1} \\ y_{2,t-1} \\ \end{pmatrix} + \begin{pmatrix} u_{1,t} \\ u_{2,t} \\ \end{pmatrix}, $$ the forecasts are $\hat y_{i,t+h|t}=y_{i,t}$ for $i=1,2$. The two are not the same.

So my question is if I estimate a VAR in first-differences, is there any problem with just applying the forecasted first-differences iteratively with an initial value equal to the last observed data point in the time-series?

No problem there. For an $h$-step-ahead forecast you may simply take the $h$ predicted differences for time periods $t+1,\dots,t+h$ and add them to the last observed value $y_t$ to obtain the forecast $\hat y_{t+h|t}$ for $y_{t+h}$ as of time $t$: $$ \hat y_{t+h|t} = y_t + \widehat{\Delta y}_{t+1|t} + \dotsc + \widehat{\Delta y}_{t+h|t} $$

Is this equivalent to taking the first-differenced VAR model (i.e. the coefficients) and applying it to the level data to obtain forecasted levels?

No. Here is a counterexample: consider an estimated model in first differences, $$ \begin{pmatrix} \Delta y_{1,t} \\ \Delta y_{2,t} \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} \begin{pmatrix} \Delta y_{1,t-1} \\ \Delta y_{2,t-1} \\ \end{pmatrix} + \begin{pmatrix} v_{1,t} \\ v_{2,t} \\ \end{pmatrix}. $$ It will produce forecasts $\widehat{\Delta y}_{i,t+h|t}=h\Delta y_{i,t}$ and thus $\hat y_{i,t+h|t}=y_{i,t}+h\Delta y_{i,t}$ for $i=1,2$.
However, when the estimated coefficients are applied on data in levels, $$ \begin{pmatrix} y_{1,t} \\ y_{2,t} \\ \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} \begin{pmatrix} y_{1,t-1} \\ y_{2,t-1} \\ \end{pmatrix} + \begin{pmatrix} u_{1,t} \\ u_{2,t} \\ \end{pmatrix}, $$ the forecasts are $\hat y_{i,t+h|t}=y_{i,t}$ for $i=1,2$. The two are not the same.