Is there a way that I can export a list of all table triggers on a server?
This will give you a list of tables that have triggers, and the name of the associated trigger:

SELECT t.name, tr.name 
FROM sys.triggers tr
    INNER JOIN sys.tables t ON tr.parent_id = t.object_id


This will give you the same list, with a column that shows the actual text of the trigger statement:

SELECT t.name, tr.name, m.definition 
FROM sys.triggers tr
    INNER JOIN sys.tables t ON tr.parent_id = t.object_id
    INNER JOIN sys.sql_modules m ON tr.object_id = m.object_id


As Aaron suggested, this would work for SQL Server 2005+ as well:

SELECT t.name, tr.name, OBJECT_DEFINITION(tr.object_id) AS TriggerText 
FROM sys.triggers tr
    INNER JOIN sys.tables t ON tr.parent_id = t.object_id


And per his excellent suggestion, the following works for SQL Server 2000+ (including SQL Server 2012):

SELECT [table] = t.name, 
    [trigger] = tr.name,
    c.text
FROM sysobjects AS t
    INNER JOIN sysobjects AS tr  ON t.id = tr.parent_obj
    INNER JOIN syscomments AS c  ON tr.id = c.id
WHERE tr.xtype = 'TR'
    AND t.xtype = 'U';