As a negative repercussions, sanctions and visa regimes may constitute a problem for multiple passports bearers - dual Iranian - or Russian - and UK citizen entering US or similar scenario that will trigger more scrutiny or even result with declined visa or access to financial services.
Naturally, if your home or original country (or one you want to visit) is at war with the country of your 2nd citizenship, you could have problems-- even "internment (prison) as an enemy alien." This is why in my PT books, etc, I always advise people to be very discreet. Do not brag about, or advertise any second citizenships. In other words, your 2nd passport is like a spare tire. Never use it for anything in your home country. If asked on a passport application if you have any other passports or citizenships, used the obvious work-around and do not reveal it, i.e--->
Use an alternate name in your second PP.
Taxation won't exist for you in your 2nd passport country if you are in that country less than (usually) six months per year . You can't have any visible taxable businesses or income property there.
Obviously you should not get on their registers by voluntarily filing any unneeded income tax returns ...
Normally just getting a local tax i.d. number is OK. Why? Because you probably need a tax number to rent or buy an apartment, car or even to get a utilities connection in most places.
If you are worried about your home country becoming dangerous or unliveable, it is a good idea to have a home in the 2nd PP country ready, waiting, and well stocked with supplies. I also advise establishing the means to support yourself for at least a year if you need to flee there. A stash of silver or gold coins is OK for this.
As to any military draft, if you are of military draft age, and INSIDE THE COUNTRY, that could be a problem. As I said elsewhere,
If your home or original country (or the place you want to visit) is at war with the country of your 2nd citizenship, you could have problems-- even "internment (prison) as an enemy alien."
This is why in my PT books, etc, I always advise people to be very discreet. Do not brag about, or advertise any second citizenship.
In other words, your 2nd passport is like a spare tire. It would be good to have if you were in Ukraine or Gaza when they became war zones. You could only get out if you had a 2nd passport. Never use it for anything in your home country until the one time in a lifetime, it becomes your key to escaping! If asked on a passport application if you have any other passports or citizenships, use the obvious workaround. Never reveal it. IMO it is best to have/use an alternate name in your second PP. If your name is Josh Williams on your home country passport, and your middle names are Harry & Quintus, your 2nd passport might be in the "unrelated" name of J. Harry Quintus. IMO You should carry a name change
certificate& birth
certificate whenever you use your 2nd PP. Why? To prove you are "one in the same person;" If that ever is necessary. Books could be written about 2nd passports and how (and how not) to use them, so don't take this brief note as more than a hint. Not legal advice!
Usually you are doing second citizenship via Investment, these are essentially shadow financial hubs, coupled with some form of boat registering, or tourism etc, so they are basically tiny countries... protected due to their location by much larger nations (Caribbean is protected by the US, then the UK/France etc).
The ones you need to worry about are citizenship via residency etc, i.e non-pay-to-play (think South America etc).