Can the spell Spirit Shroud heal a zombie? The paladin in our party cast it and was upset that the DM would not let the necrotic damage each round heal our cleric's zombie.
In 5e, necrotic energy does not heal undead
Your DM is right in their ruling. While in earlier editions of D&D, necrotic (or "negative" as it was called then) energy did heal undead, this is not generally the case in 5e.
In fact, most undead will get damaged by necrotic energy, just like any other creature. There are only a handful of powerful undead like the Lich that even have damage resistance to it. Zombies do not.
For a spell to heal undead with necrotic energy in 5e, the spell would need to explicitly say so, like the spell Negative Energy Flood from XgtE, which states:
If you target an undead with this spell, the target doesn't make a saving throw. Instead, roll 5d12. The target gains half the total as temporary hit points.
The spell description of Spirit Shroud states:
Until the spell ends, any attack you make deals 1d8 extra damage when you hit a creature within 10 feet of you. This damage is radiant, necrotic, or cold (your choice when you cast the spell). Any creature that takes this damage can’t regain hit points until the start of your next turn.
Nothing in the description of the spell references healing undead. So unless your DM decides otherwise, it won't heal them.
As far as I know, the positive/negative energy mechanics were scrapped in 5e, compared to previous editions. Any healing side effect on undead is always written in the spell's description if relevant (for example, see the spell Negative Energy Flood, from Xanathar's Guide To Everything).
In the end, your DM makes the rules
As Doppelgreener mentioned, it would seem like your DM has set up some homebrew rules to emulate positive/negative energy interactions with undead, probably inspired by previous editions. Since nothing of the sort seems to exist in the 5e official rules, it's up to them to rule whether some spells can or cannot heal undead.