What they said. Don't send condensing or near-condensing temp exhaust up a flue not designed for it, or without a system for collecting & disposing of the condensate. The condensation temp of atmospheric-drafted stack gases after the dilution vent is quite low compared to before the dilution, but not so low that it won't condense on the flue liner after you wring another 50-100F out of it with a stack heat exchanger. (The whole point of the diluting the exhaust prior to the flue is to reduce flue condensation!)
And tempering tanks in series with the HW heater only run at the efficiency of your heating system, since they pull heat from the air. In many instances that's no better than the efficiency of the HW heater (EF, and AFUE test number notwithstanding.) The savings, if any are very marginal indeed, but it will give you slightly quicker recovery and a modest increase in first-hour gallons. It's generally not worthwhile in heating dominated climates, but can be reasonable in cooling-dominated climates if local energy prices are high enough.
The amount of energy you're sending up the stack when firing is ~20% of the energy released by the fuel, the rest goes into the water. (A steady-state combustion efficiency of ~80%). And to run a stack economizer safely/efficiently you'd have to run a circulation pump to keep the temp/pressure in the heat exchanger below boiling. How much complexity/money are you willing throw at collecting MAYBE half of the stack energy if you've designed it right?
Another another 20-25% of the fuel energy is typically lost from the water via the combined standby loss of air convection through the tanks heat exchanger (not much you can do there), and the conducted losses from the near-tank plumbing and sides (about which you can do a LOT!). If you're competent to do it without restricting the dilution air at the stack or the combustion air at the burner, a retrofit HW blanket puts a big dent in the standby loss. And adding R4+ (5/8" + wall) pipe insulation on the near-tank plumbing (anything within 6' of the tank, including the pressure/temperature valve and cold water inlet plumbing) is often an even bigger cut.
Of the heat that remained in the water, typically 15% of that is lost in the homes HW distribution plumbing, mostly the abandoned BTUs of small draws of water, where you run a gallon of water to get the hot to the tap, then use a quart. Insulating the distribution plumbing to R2 (the cheap 3/8"- wall stuff found at box stores) buys you ~15-20 minutes between draws to where the water remains at a useful temp. Insulating it to R4+ buys you more. That 15% distribution loss can represent as much as a 25-30% of the fuel used to heat water.
The tank standby & distribution losses end up heating the house if it's all inside of conditioned space, so in heating-dominated climates the loss is less than in cooling dominated climates (where you pay to actively pump that heat out), but it shouldn't necessarily be ignored even in heating dominated climates.
If you're running 3 or more showers/day off this tank it may be worthwhile to recover a heluva lot more heat than any stack-economizer could by preheating the incoming water with a drainwater heat-recovery heat exchanger. (But expect to pay north of $500 for one of worthwhile size=performance.) It only works for shower draws, not baths, since the water has to flow simultaneously with the drain, but it's worth ~25-35KBTU/h while it's running, if plumbed optimally, which is as much as the output of the tank's burner itself when firing. The return is on the order of 50% during shower-length draws, so if the typical HW use is 40% for showers (as is commonly estimated), that's a 20% reduction in HW heating fuel use. If your shower use is more (as is often the case), it's an even larger fraction, and the net-benefit is greater than swapping the tank for a highest-efficiency condensing water heater(!).
But if your total HW fuel use is only 200-250therms/year, at $1/therm, it's hard to make a financial case for anything more complicated than $50 of heater blanket & pipe insulation.