Does salt affect the boiling time of water? If I have 1 cup of water on the stove and another cup of water with a teaspoon of salt. 
would the salt change the boiling time of the water?
 A: Since you are interested in time two factors have to be considered:
1) The increase in the boiling point temperature for the salt solution. This requires that more energy (aka heat) be transferred to the solution than the pure water to reach boiling. If this were the only consideration then indeed it would take more time to reach boiling point for the salt solution.
2) The change in conductivity and viscosity of the solution over pure water. To change the temperature of the fluids, heat is transferred from the pan (or beaker?) to the fluid and the presence of ions from the dissolved salt actually promote heat transfer within the fluid. This affect would tend to shorten the time. But adding salt also increases fluid viscosity. This means that convective cells within the fluid which also help distribute heat would be slower in the salt solution, so increasing time to boiling point.
All other factors being the same, it's difficult if not impossible to predict analytically using first principles which would take more time to boil. You can only accurately predict that the boiling point will be raised for the salt solution.
So the best way to make this determination is by experiment.
A: It depends on how much salt you add.
Adding salt increases the boiling temperature but decreases the heat capacity, so it takes less heat for the solution to boil. For small amounts of salt, the total time increases; for larger amounts, it decreases.
Using simple chemistry, we can estimate how much longer boiling would take. The elevation in boiling temperature caused by adding the salt can be expressed as
$$\Delta T=Kib$$
where $K\approx 0.5 \,\mathrm{K \,kg \,mol^{-1}}$ for water is called the ebullioscopic constant, $i\approx 2$ for salt ($\mathrm{NaCl}$ dissociates in water) and $b$ is the molality, that is, the number of moles of salt per mass of water.
Heat is absorbed according to the law
$$Q = M C (T_f-T_i)$$
where $M$ is the mass, $C$ is the heat capacity, and $T_i$ and $T_f$ are the initial and final temperatures, respectively.  The heat capacity of the solution of water + salt as a function of concentration (by mass) can be found here. Note that adding salt lowers the heat capacity.
Let $m_w$ be the mass of the water and $m_s$ be the mass of salt. $C_w \approx 4200\,\mathrm{J\, kg^{-1}\, K^{-1}}$ is the heat capacity of water, and $C_{sol}$ the heat capacity of the solution. To boil, the water without salt needs to absorb heat in the amount of
$$Q_1 = m_w C_w (T_0-T_i)$$
The solution of water + salt has more mass and needs to reach a higher temperature, so it needs an amount of heat equal to 
$$Q_2 = (m_w+m_s) C_{sol} (T_0+\Delta T-T_i)$$
Assuming a constant heat source, the time required to heat to a certain temperature is proportional to the amount of heat needed. So if $t_1$ is the time it takes for pure water to boil and $t_2$ is the time it takes for water + salt to boil, the ratio of these two is
$$\frac{t_2}{t_1}=\frac{Q_2}{Q_1}=\frac{C_{sol}}{C_w}\left(1 + \frac{m_s}{m_w}\right)\left(1+ \frac{\Delta T}{T_0-T_i}\right)$$
Now suppose we use $1\,\mathrm{kg}$ of water and start at $T_i=20^\circ \mathrm{C}$. Say we add $10 \,\mathrm{g}$ of salt; we can use this online calculator to obtain the number of moles, which in this case is $0.17\,\mathrm{mol}$. The changes in $C$ and $m$ are negligible for such a small amount of salt, so we get
$$\frac{t_2}{t_1}=1.01$$
Thus, it takes about $1\%$ more time if you add a small amount of salt.
Say instead we add $50\,\mathrm{g}$ of salt, corresponding to about $0.86\,\mathrm{mol}$. Then according to the source above, the heat capacity of the solution is going to be $C_{sol}\approx 3900\,\mathrm{J\, kg^{-1}\, K^{-1}}$. Therefore, we get
$$\frac{t_2}{t_1}\approx \frac{3900}{4200} \times 1.05 \times \left( 1 + \frac{0.86}{80} \right) \approx 0.985$$
So if we add a larger amount of salt, the lowering in heat capacity is sufficient to offset the increase in boiling temperature. The solution of  water + salt heats faster.
Note however that these changes are tiny. As pointed out in the comments, the non-linearity of the boiling temperature elevation law might affect the validity of results for larger concentrations. Moreover, hard-to-account-for effects caused by the change in density and viscosity might be relevant.
A: Yes and no.
It will not change the boiling time of water. If you add salt, then it's not water anymore, it is now a new solution (salt + water).  It will change the boiling point of the solution. Because that solution now has a different boiling point, if nothing else changes, it will take a different amount of time to boil.
A: If you add salt to water then put it to boil, then it would take longer to boil because the air molecules are blocked from the salt so they can't escape, and because the salt water needs more heat, it takes longer for it to boil. Technically, the more salt you add the longer it take to boil, and the less salt you add the shorter time it takes.
