How to display one value of Observable<any[]> in Angular template until next click

First of all, I'd make use of the component pattern by creating a component to display one question:

import { Component, Input, Output, EventEmitter, ChangeDetectionStrategy } from '@angular/core';

export interface IQuestion {
  label: string;
  answers: string[];
}

@Component({
  selector: 'app-question',
  template: `
    <div>
      <b>Question:</b>
      <p>{{ question.label }}</p>

      <b>Possible answers</b>
      <p *ngFor="let answer of question.answers">{{ answer }}</p>      
    </div>
  `,
  styles: [``],
  changeDetection: ChangeDetectionStrategy.OnPush
})
export class AppQuestionComponent {
  @Input() question: IQuestion;
}

Then, the AppComponent's code and its comments are enough to understand I think:
TS code

@Component({
  selector: 'my-app',
  templateUrl: './app.component.html',
  styleUrls: ['./app.component.css']
})
export class AppComponent implements OnInit {
  question$: Observable<IQuestion>;
  questions$: Observable<IQuestion[]>;

  _click$ = new Subject<void>();
  click$ = this._click$.startWith(null);

  constructor(private firebaseService: FirebaseService) { }

  ngOnInit() {
    // get the questions from firebase
    this.questions$ = this
      .firebaseService
      .getFirebaseList('your-list')
      // add a false statement so we know when to ends
      .map(questions => [...questions, false]);

    const questionsOneByOne$ = this
      .questions$
      .mergeMap(
        questions =>
          questions.map(
            // explode each question to a different stream value...
            question => Observable.of(question)
          )
      )
      // ...so we can get them one by one
      .concatAll();

    this.question$ = Observable
      .zip(questionsOneByOne$, this.click$)
      .map(([question, _]) => question);
  }

  nextQuestion() {
    this._click$.next();
  }
}

HTML code

<div *ngIf="question$ | async as question; else noMoreQuestions">
  <app-question [question]="question"></app-question>

  <button (click)="nextQuestion()">Go to next question</button>
</div>

<ng-template #noMoreQuestions>
  No more questions
</ng-template>

Here's a live demo on Stackblitz (with a mocked firebase list)
https://stackblitz.com/edit/angular-mbnscx

Let me know if you have further question(s) ;)

You should just render each item (question object) of your question array at a time if you don't want to display previous questions. I would add a new observable variable type Question for rendering to the view. An index variable to keep track of the item in array. Something like:

@Component({
      selector: 'app-quiz',
      template: `
        <md-grid-list cols="1" rowHeight="2:1">
          <md-grid-tile>
            <md-card *ngIf="question | async">
              <md-card-header>
                <md-card-title>{{(question | async)?.course}}</md-card-title>
                <md-card-subtitle>{{(question | async)?.chapter}}</md-card-subtitle>
              </md-card-header>
              <md-card-content>
                <p>{{(question | async)?.question}}</p>
              </md-card-content>
              <md-card-actions>
                <button md-button>See Answer</button>
                <button (click)="nextQuestion(index + 1)" md-button>Next 
    Question</button>            
              </md-card-actions>
            </md-card>
          </md-grid-tile>
        </md-grid-list> 
      `,
      styles: [`.example-card { width: 400px;}`]
    })

    export class QuizComponent implements OnInit {

      questions: Array<Question>;

      question: Observable<Question>;
      index: number = 0;

      constructor(private fbDatabase: FirebaseService) { }

      ngOnInit() {
          this.fbDatabase.getFirebaseList('/questions').subscribe((res) =>{
              if(res) {
                console.log(res);
                this.questions = res;
                this.question = this.nextQuestion(index);
              }
          });

      }

      nextQuestion(i: number): Observable<Question> {
          if i >= this.questions.length {
            i = 0;
          }

          return Observable.of(this.questions[i]);
      }

    }

Let me know if that works for you?