Can I drive several LEDs using a Li-Ion charging chip
Maybe, but probably not well if it works at all.  

A constant current drive is exactly what you want here, and yes, if one diode in a series string fails short the constant current driver will adjust to maintain that current even with a much lower forward voltage.  

However, battery charging is not as simple as providing a constant current into the cell, there are voltage and time thresholds that have to be observed as well.  The TP4056's datasheet (at least the versions I've seen) do not describe its behavior in tremendous detail, but what it does show is that its behavior changes depending on if the battery voltage is &lt;2.9V, &lt; 4.2V, or >4.2V.  Only in that middle range does the TP4056 provide the programmed constant current.  The charger may not even begin charging if it does not detect a connected battery, this is a quite common and reasonable limitation, although I don't know if the TP4056 has this behavior.  It might take some tricks to get it to even turn your LEDs on at all.

Finally, Since the TP4056 is an incredibly common part I would not be surprised if many different manufacturers make equivalents, and these may have subtly (or drastically) different behaviors, especially when used outside of their intended purpose, so even if you can get it to work with one particular chip there's no guarantee that the next ones you buy will work as well (if at all).

On the whole there are much better options, from a basic constant current source made from a single transistor and resistor to a full-blown buck-boost converter and everywhere in between.  There are even LED driver ICs that are designed for battery supplies and have useful features built in.  All of these will likely be better, on the whole, than using the TP4056.