You'll get different answers. In principle, if you get things hot enough most of the typical disease pathogens will be destroyed during composting. That said, if I suspect diseased plant material, it doesn't go into my compost. It's a risk I personally try to avoid. Others will disagree.

Yes,no,maybe so,certainly it's ok depending on where the compost will be used and the composting process.
The cuke plants my harbor pests which can survive cold composting. A cold compost pile is a perfect place for insects to overwinter. Plants can also be deseased without dieing. I strive for a hot pile during at least the early part of composting veggie materials.

I seem to recall that tomato blight lives in the soil, so it's going to be there anyway regardless of what you do with the clippings. It's good to move tomatoes each year since they're infected from soil splashing up on the bottom leaves.

I've had horrible problems with cuke beetles, vine borers and squash bugs, so I'd be inclined to discard those plants somewhere else than my compost (i.e. trash) to get rid of eggs.

Technically, yes (for reasons given above)...realistically, I wouldn't chance it.

The small amount of plant matter that eventually becomes compost generally isn't worth the gamble. A lot depends on how good of a "pile master" you are. Personally, I don't chance it.

I just bury it in beds that will never see a tomato or squash (they are too shady for that).

There is some research that suggests that if your compost reaches temperatures in the 150 degree range disease pathogens can be destroyed, so if you can get your compost to get to those temperatures you can compost diseased plant parts.
Simply because plant parts have turned yellow does not mean they are diseased, however. There are a number of reasons why plant parts loose the chlorophyll that makes them green.

tox - the fungus that causes early blight on tomatoes (and other solanaceae or nightshades) will overwinter on plant debris, including seeds, and weeds from the about anything in the solanaceae family, as well as the soil for at least a year where those plants have grown. The spores can be transmitted to new plants by soil splashing as well as being windblown. So yeah, if it was there before it will be there in the future. The problem with recycling infected plant material is increasing the spore count and the probability of future infections.

Linked below is a decent reference titled "Minimizing Diseases in Vegetable Gardens" from the good folks at Cornell. How to handle infected plant material, including the composting option, is covered under the section "Clean up debris".

Here is a link that might be useful: Cornell - Minimizing Diseases in Vegetable Gardens

A lot of good responses here. Let me phrase this a different way.....

Generally speaking, what are the safest materials to compost?

By safest I mean the least likely to introduce organisms/toxins that would compromise the resulting compost.

On the same subject, how soon should one be able to compost lawn clippings after spot treating the lawn for weeds with a chemical herbicide (since the product would be absorbed foliarly). I would assume that for granular products (herbicide, fungicide, insectide) the lawn clippings would be fine after a decent rainstorm or two since the granules make their way into the soil.

Thanks TX for the blight info. I think I have late blight, starts in earnest about now.

nj, just about any healthy plant material is OK to compost. Plant toxins are generally not an issue, the list of things (like juglone from walnut trees) that should be kept out is waaay shorter than the list of things that are OK to put in.

As far as herbicides, it all depends on what you are using. I'm not sure you can depend on a granular product not to be absorbed into foliage and end up in clippings. However, generally speaking, the stuff you can buy at the store will break down in the compost bin. Don't use treated clippings directly on the garden as mulch, especially from a recently sprayed lawn. That's probably the biggest risk.

TXEB is the pesticide authority and will hopefully straighten me out if there's a hole in any of that.

tox - in my experience fungal diseases both in the veg garden and in the lawn seem to be greatest at seasonal turns when the days are reasonably warm -- mid 70's to low 80's -- and the night temps are breaking down into the 50's.

nj - I agree with tox. Material from about any reasonably healthy garden plant, clean grass clippings, and anything from the kitchen you are willing to put into your compost. The most common problem for most folks is getting enough brown material for the amount of greens they can produce from their kitchen in an ongoing basis, let alone fresh grass clippings.

On the lawn treatments, the two big ones for compost and plant health are herbicides and fungicides.

The issue on herbicides is carryover via your compost to your garden and subsequent killing of your plants. The transfer to the compost comes via amounts that have been taken up or absorbed by the plants treated. Most of the common herbicides available to the home owner work via foliar absorption. Only a few, mostly atrazine and Image, work primarily through translocation from the soil. If you're doing the application with common homeowner retail herbicides that you bought from someplace like HD, then they will decompose to very low residual levels under any reasonable composting conditions. There is a risk, but it is very small. To minimize that risk I suggest that after application of any herbicide you wait until after 4 regular mowings of 1/3rd grass height or greater. After that the residual in the grass should be low enough that even with marginal composting you should be without issue.

Fungicides present a different threat, and that is to the microbial community that is working in your compost. Honestly, I don't have a good feel for how long they can persist and at what levels in grass clippings. It would probably depend on whether they work topically or systemically. I don't think there is much data on this - just a few specific lab studies on or in media to determine susceptibility. I've not heard of it ever being an actual problem.

Commonly available insecticides generally aren't a concern to plant or compost health, but they could pose a risk if they are systemic in action and transfer through the compost. Again, this is one of those I don't really know about, and I don't think there is much data on that either.

A final word of caution - all present a risk from handling treated materials.

My SWAG is that grass clippings from any homeowner treated lawn would be safe to handle and compost after the 4th mowing, provided that the materials were used in accordance with the label instructions.

After all that on grass clipping, I am not a fan of composting them. I've done it in the past, but gave it up several decades ago. The best place for those grass clippings is right back on the lawn from where they came. You lawn will benefit from tremendously over time from mulch mowing and allowing the clippings to decompose in place, providing organic matter for the soil beneath the surface. That also eliminates all the risk of lawn chemical transfer to your compost system.