Short answer is: yes.
There are mitigating factors though, of course! If you are in pain or your workout of choice is crossfit or some form of boxing or martial arts that is going to make impact a necessity, consider a DIFFERENT workout while you're in the healing stages. Depending on the area you've had pierced, this can be 6 weeks to 6 months.
This week, I had the same piercing done as fashion icon and serious workout warriors, Rihanna, Kendall Jenner, Janet Jackson and Bella Hadid.
It is requiring some modification as far as NOT lying on my chest (goodbye to Child's Pose for a few weeks) but I'm still working out daily. I asked the fabulous and entirely professional and experienced Sammah Chehade (Hairhouse Warehouse, Northland) to give me the rundown on piercing after care for those who plan to workout.
1. The usual healing time on nipple piercings range roughly anywhere between 6 weeks-6 months, sometimes a little longer depending on how well they're being taken care of it and how fast your own body likes to heal itself.
2. Touching your fresh piercing, bumping/knocking it, playing with the jewellery and not cleaning it regularly can cause an infection and/or really irritate the pierced area making the piercing uncomfortable and quite sore, it will also extend the healing period. Your aim is to totally leave your piercing alone and make sure you're cleaning it when its needed!
3. It is best to avoid chlorine water and sauna's for at least the first 2-4 weeks whilst your piercing is healing, as these can irritate or cause an infection for most people. If you can not avoid it, it is recommended to have a shower straight after, rinsing the piercing out really well and then giving it a clean with your aftercare solution.
4. In terms of clothing, its best to avoid net/wool tops while healing as these can get caught and pull on your jewellery, also always wear a clean top/bra while it is healing to avoid any cross-contamination. For ladies I would recommend to wear a full cupped padded bra, any low cut bra's can knock your piercing as you move around and the padding will protect your piercing if you accidentally knock/bump it as the impact wont be as hard against the pierced area. If at any time your piercing starts to bleed again, make sure you cover it with a dressing, not leaving it exposed to bacteria to avoid infection.
5. Working out and sweating is fine! Just be careful not to knock it around as you're working out and avoid any workout that involves pressure/leaning against your chest till it heals. For ladies while working out, wear a firm padded sports bra to avoid as much movement as possible for the piercing so you don't irritate it. After a workout, have a shower and rinse out your piercing from any sweat/dirt and give it a clean with your aftercare solution 
6. The only position not recommended for your fresh nipple piercing is laying on your chest, as this can apply pressure and irritate your piercing, leaving it uncomfortable and sore! Everything else is fine although if you feel like you're going to be in a position that's going to be leaning/pressing on your piercing, avoid it!
7. You will only feel pain if you're not taking care of your piercing or touching it in any way till it is healed. You will know your piercing is fully healed when you are able to touch it, lay on it, do weights etc and it doesn't hurt! In terms of sensitivity is varies. In my piercing career, I have noticed with most males it tends to stay sensitive (in a good way!) after it is pierced and healed, With women its different, a small amount of women become more sensitive and others don't feel a difference at all once it's healed!
8. A line to I like to tell my customers or anyone thinking of getting their nipple pierced is: never let the fear of pain stop you from getting what you want! at the end of the day the piercing pain is for a second for something that could last you a lifetime! You will be proud of yourself for putting aside your fear and actually getting something you really want! but hey, what's beauty without pain 
