No, pruning off the old, dead flowers (that have become seed pods by now) won’t hurt the lilac. A better time to do this would be right after the flowers finish up in spring. This prevents the lilac from putting any energy into producing seeds and that energy will then be put into growth and more flowers the following season. Since we know that you have an old large, well-established lilac, it may be time to rejuvenate it with some heavy pruning. You’re going to need a pruning saw for this. You probably have some very large, older trunks coming out of the middle of the lilac that are not producing many flowers. These older trunks may be 10 or 20 years old and no longer have the vitality to produce flowers the way they used to. Get in there and cut them off as close to the base as possible. Make sure that you don’t remove more than 1/3 of the total branch structure to avoid shocking the plant. Remove all these older trunks and the energy that they were using will now go into younger shoots that will produce the most flowers. Ideally you should have done this in spring right after the lilac bloomed. Lilacs form buds in the summer after flowering. Those latent buds winter over and produce flowers in spring. If you had pruned the lilac this spring, you’d have lots of new growth with buds ready to go. There’s nothing wrong with cutting your lilac back now but you won’t get the big benefit until the spring after next. People are often shy about pruning heavily but, as in this case, it is the best way to get the most out of your lilac. Once you remove the old, unproductive trunks, it will be several year before you need to do anything more than light pruning.