Welcome to the Forum!! I'm sure that others with much experience in tubes will help you with suggestions on tweaking the Eico.
On the Heresy's, there are several things that you can do to get the best out of them. There are two types - removable rear panel (H-I's) and MDF non-removable rear panel (H-II's and III's). If they are the new H-III's, you don't need to do anything other than enjoy them.
On the older H-II's and in particular the H-I's, you should consider changing the capacitors on the crossover networks. You can either do it yourself, or contact some of the Forum members. Bob Crites (BEC) or DeanG come to mind.
If you have Heresy-I's, you should carefully remove the rear panel and "seal" it with closed cell foam speaker gasket tape (parts express has it). Reason, the Heresy is an acoustic suspension speaker and any "leaks tend to hurt the bass response. Also with the rear panel off, would be a good time to remove the crossovers for the cap replacements and check the soldering connections on the drivers, etc. Also, check the torque on the screws that hold the drivers to the "motor board". I would suggest to lay the speaker down on it's front and loosen all of the driver screws and retighten them carefully and evenly, a little at a time, and by going back and forth to make sure they are "pulling" the flange of the driver down evenly. You can also add a dab of GE black silicone around the edge of each screw hole on the woofer driver to prevent "leaks". The silicone is removable if you need to replace the woofer at some point. You could also check the driver resistance with a multimeter and see what they are showing for resistance in ohms. Each "pair" should be more or less within ~0.1 - ~0.2 of each other. If they are not close to each other in measurement there there is a problem. Check with BEC for the desired values for the drivers. For midrange and tweeters, the most common problem is a bad diaphragm. Look at BEC's web site (critesspeakers.com) and you can see the different diaphragm replacement options. Woofers bad? Just replace them. Klipsch parts has the K-24's which are the replacement woofers for the H-I and H-II. You can also replace the woofers on both with the K-28 which is the new version used on the H-III.
On the Heresy-II's, check the driver screws on the front, and you might find several are not tight. In order to change the capacitors, you must remove the woofer, then the midrange, and then the tweeter. use a piece of tape to mark each wire pair, and carefully unclip the spade connectors on the drivers as you lift them out. Check the gaskets. Generally while they are re-useable, a new set from Klipsch parts are inexpensive, or you can make your own from the parts express speaker gasket tape. With the drivers out, unscrew the crossover terminal cup on the back, lift it out (it's a tight fit....) and either send it off to whomever will do the caps for you or if you are good with a soldring iron, go for it. Because it's on a printed circuit board, I generally would ask for someone else to do it. Other than the driver gaskets on the H-II, it's generally well sealed. That being said, however, you can lay down a thick bead of woodworkers glue on each internal seam and thus make sure there are no "leaks". Check the driver resistance with a multimeter. And, the same thing about even torquing the screws when "bolting" them back on. Reverse order, and make sure the spade clips are on snug. Use a pencil eraser on the spade lugs on the drivers to clean off any oxidation.
It's a good maintenance schedule every few years and will keep them up and running.
Best of luck and again, welcome!!
In the kingdom of the blind, the one eyed man is still just the bus driver....