- For most tasks, differences in participants' computers' processing speeds will not matter much. Computers these days are all quite powerful, and even on the low end the vast majority of tasks will not exhaust a given computer's resources (see e.g.
http://link.springer.com/article/10.3758%2Fs13428-014-0471-1 ).
- Technical factors *may* come into play if a specific task requires very specific technical parameters in order to be valid. For example, a task where you have participants judge absolute color values would not be suitable for being administered on their home computers which are not under your control. The validity of such a task would hinge on the computer displays all being calibrated identically, which is not something you can ensure outside of a lab environment where the measurement apparatus is fully under your control.
- The variability in technical terms between participants' home computers is unlikely to be greater than the variability between computers in, say, different labs across the world.
- Other non-technical factors are likely to play a larger role, such as the fact that you have no control over the participant's environment (noise level, other distractions, no way to clarify instructions, etc.). How malleable a specific task is to such variations is not something that can be answered generally and will likely vary with the specific task demands (e.g. tasks that demand sustained attention or complex operations over a long period of time vs. tasks that require relatively simple judgments or decisions).
- The published literature on classical paper & pencil or lab tests vs. their online counterparts largely shows they produce comparable results.
A lot has been written and published about the advantages as well as potential pitfalls of online experimentation by authors such as Reips. There also are quite a number of studies comparing online vs. lab administrations of a given experiment (a fairly recent example is
http://www.ppgia.pucpr.br/ismir2013/wp-content/uploads/2013/09/59_Paper.pdf ).
Hope this helps.