There is a way to achieve what you need and for that you need to patch your objects. You need to use the patch type merge-patch+json and as a patch you need to supply a complete/desired list of env vars.

As an example lets consider this deployment:

apiVersion: apps/v1
kind: Deployment
metadata:
    name: mydeployment
spec:
    replicas: 2
    selector:
    matchLabels:
        app: sample
    template:
    metadata:
        labels:
        app: sample
    spec:
        containers:
        - name: patch-demo-ctr
        image: gcr.io/google-samples/node-hello:1.0 
        env:
        - name: VAR1
            value: "Hello, I'm VAR1"
        - name: VAR2
            value: "Hey, VAR2 here. Don't kill me!"

$ kubectl get pods
NAME                           READY   STATUS    RESTARTS   AGE
mydeployment-db84d9bcc-jg8cb   1/1     Running   0          28s
mydeployment-db84d9bcc-mnf4s   1/1     Running   0          28s

$ kubectl exec -ti mydeployment-db84d9bcc-jg8cb -- env | grep VAR
VAR1=Hello, I'm VAR1
VAR2=Hey, VAR2 here. Don't kill me!

Now, to remove VAR2 we have to export our yaml deployment:

$ kubectl get deployments mydeployment -o yaml --export > patch-file.yaml

Edit this file and remove VAR2 entry:

apiVersion: extensions/v1beta1
kind: Deployment
metadata:
spec:
  progressDeadlineSeconds: 600
  replicas: 2
  revisionHistoryLimit: 10
  selector:
    matchLabels:
      app: sample
  strategy:
    rollingUpdate:
      maxSurge: 25%
      maxUnavailable: 25%
    type: RollingUpdate
  template:
    metadata:
      creationTimestamp: null
      labels:
        app: sample
    spec:
      containers:
      - env:
        - name: VAR1
          value: Hello, I'm VAR1
        image: gcr.io/google-samples/node-hello:1.0
        imagePullPolicy: IfNotPresent
        name: patch-demo-ctr
        resources: {}
        terminationMessagePath: /dev/termination-log
        terminationMessagePolicy: File
      dnsPolicy: ClusterFirst
      restartPolicy: Always
      schedulerName: default-scheduler
      securityContext: {}
      terminationGracePeriodSeconds: 30
status: {}

Now we need to patch it with the following command:

$ kubectl patch deployments mydeployment --type merge --patch "$(cat patch-file.yaml)"
deployment.extensions/mydeployment patched

Great, If we check our pods we can see that we have 2 new pods and the old ones are being terminated:

$ kubectl get pods
NAME                           READY   STATUS        RESTARTS   AGE
mydeployment-8484d6887-dvdnc   1/1     Running       0          5s
mydeployment-8484d6887-xzkhb   1/1     Running       0          3s
mydeployment-db84d9bcc-jg8cb   1/1     Terminating   0          5m33s
mydeployment-db84d9bcc-mnf4s   1/1     Terminating   0          5m33s

Now, if we check the new pods, we can see they have only VAR1:

$ kubectl exec -ti mydeployment-8484d6887-dvdnc -- env | grep VAR
VAR1=Hello, I'm VAR1