Stiff equations tend to use "implicit" methods. I am actually working on an implicit simulation (rigid body constraints) which is infinitely stiff! Implicit methods are basically guess and check: try to find a set of forces that satisfies future constraints. This is a well-developed field, multigrid conjugate-gradient based methods are among the state of the art.
So how would you use an implicit method? Take a pendulum for which we have a single anchor-spring-mass as an example. The spring is stiff, almost a rigid rod, and you don't care about the high-frequency shortening and lengthening vibrations of the spring. You do care about modeling the overall swinging motion, however. "Normal" (explicit) solvers need a time-step that is small enough to capture the spring's vibrations. For an implicit scheme, the step is set by the dynamics we care about: the swing period of the pendulum, which is much longer than the vibrational period of the spring. Thus implicit methods, although more expensive per step, allow us to gloss over high-frequency degrees of freedom that we don't care about (as long as they these DOF's are nearly linear, as is true in our case with small spring vibrations).
For the pendulum case it is easy to create a new model bases on an ideal rigid rod, eliminating the annoying high-frequency DOF. For more complex cases this is difficult to do, however.
Stiff equations in chemistry typically arise when reaction time-constants are small. In a first order decay reaction: dC/dt = -k*C, the time constant is 1/k. For non-linear complex reactions, you can look at small perturbations of concentration and see how quickly they grow/decay.
Another way is for diffusivity to be high. The time constant is ~ neighbor distance^2/(diffusivity). It shrinks rapidly with decreased particle size.
Chemical stiffness is an issue if these time-constants are smaller than the SPH time-step needed. In this case you can decouple the simulations: take an SPH step and take several smaller diffusion and/or reaction sub-steps, or use an implicit model for the diffusion/reaction. Implicit methods for diffusion work well since error tends to blurs out, as long as total mass is conserved.
If the only stiff aspect is reaction kinetics, here's a simple way to use "sub-steps": for each SPH step calculate the diffusion fluxes. Assume a constant flux into your particle over the entire SPH timestep. For your particle, calculate reaction kinetics using much smaller sub-steps until you found your concentrations at the end of the SPH timestep. This way you capture the fast kinetics without any extra expensive SPH calculations.
SPH simulations themselves get stiff when fluids become incompressible and/or resolution gets higher. The step size ~ (particle size)/(speed of sound), the latter depends on the "spring constants" in the pressure terms and how many neighbors particles have, as well as the mass of the particles. Both higher speed of sound and lower fluid velocities make a more incompressible fluid, and both both cause "stiffness" in the sense that they increase how many steps are needed to for the dynamics you are interested in happen.