That's a great way to make it not be used for real things
Trust is a key component among employees. Letting your investors into that mix will just lower the trust there, and get less useful stuff on the todo list.
If they do not trust you, it is possible that they view you as some kind of glorified employees and see themselves as your boss. They're not, they're just investors. It might be worth to give this relationship some thought and then discuss it with them somehow. You didn't start a startup to gain another meddling boss.
When you say "investors", do you mean "potential investors we are currently courting", or "shareholders"?
If you mean "shareholders", the answer is "no". If they have a seat on the Board, then they are entitled to board-level information, but this wouldn't generally involve unfiltered access to your task list.
If you mean "potential investors", then the next question is: "as a one-time thing, or on an ongoing basis?" If you mean "on an ongoing basis", the answer is "no"-- there's no reason to give someone on the outside this kind of information regularly. If, however, you mean "as a one-time thing", then the answer could be "yes", as it might be a good way to give the investors a better view of what you are trying to achieve, and where you are in the process.
So I would say show them before they invest and see what happens.
2. If you are hard working and having good progress, it should not matter too much.
95% of the information shared in an organization and put onto a task list is about milestones that need to be hit, progress towards those milestones, and information that needs to be gathered to make this happen. All of this benefits from more helpful eyes on it. If your investor happens to know someone at an organization that your salesperson is trying to sell then they should help. If they have another portfolio company solving similar engineering problems, your teams should get together.
Most people are intelligent enough to be able to use this information to help and not try and micromanage. If they do try to do that, then you tell them to lay off and stick to helping people do their jobs instead of interfering.
Put it another way -- if you don't think your investors can handle this extra information and use it to be helpful, why in the world did you take money from them?
While I've never heard of investors getting involved down to this level of detail, perhaps in certain circumstances it could be beneficial for them to offer guidance on accomplishing certain tasks if they have the skillset to do so. I tend to doubt that anything good would come of this idea though.
The purpose is for them not to meddle (eek!) but to be more informed, (give better suggestions, know how close you are to a milestone, have less meeting)
For the team, itll be a good motivational tool and reinforcement mechanism.
ITs a bad idea if you have a bad investor, good idea if you have a good investor.
---end of devil advocation (?)