It looks like an issue with the Trigger Code, it can be a code quality issue (SOQL Query in a loop, excessive use of SOQL, etc. ).
I think that you can still load your data but a code review should be done to bulkify the triggers code.
If the Business logic implemented in the trigger (on that object you're loading) can be bypassed, you can deactivate the trigger by unchecking the "active" checkbox. This is cannot be done directly in the Production environment.
Either by deployment: Redeploy Triggers with unchecked Active checkbox
OR by Implementing a bypass mechanism in Apex
Another easy approach is to reduce the batch size in Data Loader to a low value where the limit issue is not encountered (I know it will take an eternity to load you but can try decreasing the batch size gradually 50 then 20 then 10 or even 1 - loading 1 by 1 will be really long).