What happens to my ticket from London to Paris if there is a labor strike going on?
Can we still get there?
Do we need to reschedule?
Hard to say in general.
Not all strikes stop all trains. Some will stop only local trains or only international trains. Other strikes will be more general and make all train travel impossible.
Mostly you can only be sure on the day, as mostly the train companies only know on the day whether they have to cancel the trains.
If you have the flexibility to reschedule to a day without strikes, you may as well do so.
If you have to travel on the day, you can look into other methods of travel, like coach or crossing the sea by ferry. But if you decide not to use your ticket, do not expect a refund unless the train was really cancelled. Best call the train company before you decide. If a strike is announced, they might change your ticket to one for an other day.
The Eurostar website has information on how to claim your money, read the page carefully, as some cases will only give you a reduction on a new ticket while other may give a full refund.
The Eurostar website also has a link for actual trains running, which is the top of the list on the left on this page.
If you need additional train travel, either in the UK or in France you can find the information on the websites for those rail companies.
This page for France (does include Eurostar as well.)
I can not find a UK strike information page right now, but if you enter your planned trains into the live departures on this page you should get information if they are influenced by strikes.
For the mid-May 2016 strikes, you're in luck. As detailed on the Eurostar status update page:
Planned strike action in France from Tuesday 17th of May 20.00 CET to Thursday 19th of May 2016 20.00 CET.
... Onward connections within France may also be affected by this action, and we would encourage all customers to check onward travel plans with the relevant transport provider as soon as possible.
... Eurostar services will run as normal.
However, it does vary between strikes. When there was a Belgian rail strike a few years ago, Eurostar terminated all their Brussels services in Lille.
Oh, and see our Is there a way to look up live delay information for Eurostars? question for more on looking up realtime Eurostar information!