Economic theory provides a number of useful tools for thinking about tariffs, and these tend to frame economists' instincts when it comes to discussions about "free trade agreements." However, in many cases, tariffs are already quite low (perhaps this is a rare success for economists' powers of persuasion...) and the main ingredients of trade agreements concern other things which are trickier to analyze.
One aspect of contemporary trade agreements that is coming under scrutiny in the discussions over the Transatlantic Trade and Investment Partnership (TTIP) and the Trans-Pacific Partnership (TPP) are provisions to protect foreign investors by allowing them to take disputes with governments to arbitration. This Vox piece by Danielle Kurtzleben is a nice summary of the debate concerning these investor-state dispute settlement (ISDS) rules.
Another issue getting considerable attention in the TPP discussions is the fact that trade agreements typically do not deal with currencies. As the Times reported, many in Congress are pushing for incorporating a provision to deal with "currency manipulation" into the TPP.
This is a tricky issue which cuts across economics' division between international trade - which uses microeconomic theory to analyze long-run equilibria - and open-economy macroeconomics, which is concerned with monetary and balance of payments issues (which are "short-run" but can have meaningfully persistent effects). At an institutional level, trade policy is usually the purview of trade ministers (e.g., the US Trade Representative), while currency policy falls to central banks and finance ministers (i.e., the Treasury in the US). Globally, trade has the WTO, while currencies have the IMF (which, unlike the WTO, does not have any enforcement mechanisms).
Simon Johnson and Jared Bernstein have written in favor of inserting a currency clause, while Edwin Truman argues the contrary. Janet Yellen expressed concern about the potential for trade agreements to encroach on monetary policy, and Jeffrey Frankel noted that some of the loudest concerns about currency manipulation aimed at China (not currently a party to the TPP) are out of date.