The question about the availability of closed-form solutions can generally not be answered for types of options alone but only for the combination of a payoff function and the underlying asset dynamics.
Consider for example European plain vanilla options. These have a (quasi) closed-form solution when the underlying follows an exponential Levy processes including geometric Brownian motion (GBM), in many stochastic volatility models but not under local volatility dynamics. However, you will usually find that options that cannot be priced in closed-form under GBM cannot be priced in closed-form under other (sensible) underlying asset dynamics either.
Furthermore, many more exotic options that can be priced in closed-form under GBM should not be priced using this model. The reason is that the value of the payoff is heavily driven by risk-factors whose dynamics are not modelled. Examples are the forward skew sensitivity of American binary options, the stochastic interest rate exposure of equity linked bonds with an uncertain maturity, ... .
Once the option cannot be priced in closed-form under the respective model dynamics, you need a numerical method such as finite differences, Monte Carlo simulation, trees, ... .